Lawvere Categories as Composed PROPs

. PROPs and Lawvere theories are related notions adapted to the study of algebraic structures borne by an object in a category, but whereas PROPs are symmetric monoidal, Lawvere theories are cartesian. This paper formulates the connection between the two notions using Lack’s technique for composing PROPs via distributive laws. We show Lawvere theories can be seen as resulting from a distributive law of two PROPs — one expressing the algebraic structure in linear form and the other expressing the ability of copying and discarding variables.


Introduction
PROPs [28] are symmetric monoidal categories with objects the natural numbers. In the last two decades, they have become increasingly popular as an environment for the study of diagrammatic formalisms from diverse branches of science in a compositional, resource sensitive fashion. Focussing on computer science, they have recently featured in algebraic approaches to Petri nets [11,35], bigraphs [12], quantum circuits [15], and signal flow graphs [5,7,1,20].
PROPs describe both the syntax and the semantics of diagrams, with the interpretation expressed as a PROP morphism [[·]] : Syntax → Semantics. Typically, Syntax is freely generated by a signature Σ of operations with arbitrary arity/coarity and can be composed sequentially and in parallel. Thus diagram syntax-which we refer to as Σ-terms-is inherently 2-dimensional: the term structure is that of directed acyclic graphs, rather than trees, as in the familiar case of operations with coarity 1. A crucial aspect is linearity: variables in Σ-terms cannot be copied nor discarded.
It is often useful to axiomatise the equivalence induced by [[·]] by means of a set of equations E, and then study the theory (Σ, E). For PROPs, completeness proofs typically provide a serious challenge, involving the retrieval of a normal form for Σ-terms modulo E. The difficulty can be drastically reduced by exploiting certain operations on PROPs: an example of this modular methodology is provided by [5,7], where the PROP operations of sum and composition are crucial for giving a sound and complete axiomatization of signal flow diagrams.
Sum is just the coproduct in the category of PROPs. Whenever two PROPs T 1 and T 2 can be presented by the theories (Σ 1 , E 1 ) and (Σ 2 , E 2 ), then their sum T 1 + T 2 is presented by the disjoint union (Σ 1 Σ 2 , E 1 E 2 ).
Composition of PROPs is more subtle, as it requires certain compatibility conditions between the structure of T 1 and T 2 . Lack [25] describes this operation formally by means of distributive laws, seeing PROPs as monads in the 2-categorical sense of Street [36]. In a nutshell, a distributive law λ : T 1 ; T 2 → T 2 ; T 1 of PROPs is a recipe for moving arrows of T 1 past those of T 2 . The resulting PROP T 2 ; T 1 enjoys a factorisation property: every arrow in T 2 ; T 1 decomposes as one of T 2 followed by one of T 1 . The graph of λ can be seen as a set of directed equations E λ := ( −→) and T 2 ; T 1 is presented by the theory (Σ 1 Σ 2 , E 1 E 2 E λ ).
This work uses distributive laws of PROPs to characterise Lawvere categories 4 , a well known class of structures adapted to the study of categorical universal algebra. The essential difference with PROPs is that Lawvere categories express cartesian theories (Σ, E), i.e. where Σ only features operations with coarity 1 and E may include non-linear equations. Our starting observation is that the Lawvere category L Σ on a cartesian signature Σ exhibits a factorisation property analogous to the one of composed PROPs: arrows can always be decomposed as where Cm is the PROP of commutative comonoids, generated by a copy 1 → 2 and a discard 1 → 0 operation, and T Σ is the PROP freely generated by Σ. This factorisation represents cartesian Σterms by their syntactic tree -the T Σ -part -with the possibility of explicitly indicating variable-sharing among sub-terms -the Cm-part. This simple observation leads us to the main result of the paper: for any cartesian signature Σ, there is a distributive law of PROPs λ : T Σ ; Cm → Cm ; T Σ which is presented by equations that express the naturality of copier and discarder; the resulting composed PROP Cm ; T Σ is the Lawvere category L Σ .
By a quotient construction on distributive laws, it follows immediately that the above theorem holds more generally for any cartesian theory (Σ, E) where the set of axioms E only contains linear equations. For instance the Lawvere category of commutative monoids L Mn can be obtained by means of PROP composition, while the one L Gr of abelian groups cannot, because of the nonlinear axiom x × x −1 = 1. Obviously, one can still formulate L Gr as the quotient of the composite L Mn by adding this equation, see Example 4.6 below.
As a side remark, we observe that, by taking the sum Cm + T Σ , rather than the composition, we are able to capture a different, well-known representation for cartesian Σ-terms, namely term graphs, which are acyclic graphs labeled over Σ. With respect to the standard tree representation, the benefit of term graphs is that the sharing of any common sub-term can be represented explicitly, making them particularly appealing for efficient rewriting algorithms, see e.g. [34] for a survey on the subject. As shown in [16], Σ-term graphs are in 1-1 correspondence with the arrows of the free gs-monoidal category generated by Σ, a concept that actually amounts to forming the sum of PROPs Cm + T Σ . Thus the only difference between term graphs and the representation of terms given by Cm ; T Σ is in the validity of naturality of copier and discarder. Intuitively, a term where a resource is explicitly copied is not identified with the term where two copies appear separately: in short, copying is not natural.
Related Works. The motto "cartesian terms = linear terms + copying and discarding" inspired several papers exploiting the role of Cm in Lawvere categories, see e.g. [14,26,16]. In our work, Lawvere categories feature as a distinguished example of a construction, PROP composition, that is increasingly important in many recent research threads [6,5,20,32]. The significance of this exercise is two-fold. First, it gives a deeper understanding of the nature of Lawvere categories and how they formally relate to PROPs, by showing the provenance of the natural copy-discard structure. Second, our result provides a canonical means of defining a distributive law for freely generated PROPs, showing that the result of composition is a familiar algebraic notion and enjoys a finite axiomatisation.
The following construction, reported by Baez in [2], is close in spirit to our work. There is pseudo-adjunction between symmetric monoidal and categories with finite products where R is the evident forgetful functor and L adds to any object of C ∈ SMCat a natural copy-discard structure: natural diagonals and projections. Baez [2] states an equivalence between RL(C) and C ⊗ Cm, with the tensor ⊗ defined by . Indeed, our main construction, as well as being a distributive law, is also an instance of a tensor or Kronecker product of symmetric monoidal theories; a concept that has been explored in some detail in the cartesian setting of Lawvere theories, see e.g. [22]. Our work restricts attention to PROPs T Σ ∈ SMCat freely generated by a cartesian signature Σ: in this case, it is enough to add a copy-discard structure for the object 1 and RL(T Σ ) coincides with PROP composition Cm ; T Σ . Our perspective exploiting distributive laws has the advantage of providing a finite presentation in terms of the naturality axioms.
It is also worth mentioning that the relationship between symmetric monoidal and cartesian structures is central in the categorical semantics of linear logic; in this perspective, the presence of Cm allows to interpret the structural rules of contraction and weakening -see e.g. [24,30].
Prerequisites and notation. We assume familiarity with the basics of category theory (see e.g. [29,10]), the definition of symmetric strict monoidal category [29,33] (often abbreviated as SMC) and of bicategory [10,3]. We write f ; g : a → c for composition of f : a → b and g : b → c in a category C, and C[a, b] for the hom-set of arrows a → b. It will be sometimes convenient to indicate an arrow f : a → b of C as x f ∈C − −− → y or also ∈C − − →, if names are immaterial. For C an SMC, ⊕ is its monoidal product, with unit object I, and σ a,b : a ⊕ b → b ⊕ a is the symmetry associated with a, b ∈ C.

PROPs
Our exposition is founded on symmetric monoidal theories: specifications for algebraic structures borne by objects in a symmetric monoidal category.
Definition 2.1. A (one-sorted) symmetric monoidal theory (SMT) is a pair (Σ, E) consisting of a signature Σ and a set of equations E. The signature Σ is a set of generators o : n → m with arity n and coarity m. The set of Σterms is obtained by composing generators in Σ, the unit id : 1 → 1 and the symmetry σ 1,1 : 2 → 2 with ; and ⊕. This is a purely formal process: given Σterms t : k → l, u : l → m, v : m → n, one constructs new Σ-terms t ; u : k → m and t ⊕ v : k + n → l + n. The set E of equations contains pairs (t, t : n → m) of Σ-terms with the same arity and coarity.
The categorical concept associated with symmetric monoidal theories is the notion of PROP (product and permutation category [28]).

Definition 2.2.
A PROP is a symmetric strict monoidal category with objects the natural numbers, where ⊕ on objects is addition. Morphisms between PROPs are strict symmetric identity-on-objects monoidal functors: PROPs and their morphisms form the category PROP. We call a sub-PROP a sub-category of a PROP which is also a PROP; i.e. the inclusion functor is a PROP morphism.
The PROP T freely generated by an SMT (Σ, E) has as its set of arrows n → m the set of Σ-terms n → m taken modulo the laws of symmetric strict monoidal categories - Fig. 1 -and the smallest congruence (with respect to ; and ⊕) containing the equations t = t for any (t, t ) ∈ E.
There is a natural graphical representation for arrows of a PROP as string diagrams, which we now sketch, referring to [33] for the details. A Σ-term n → m is pictured as a box with n ports on the left and m ports on the right. Composition via ; and ⊕ are rendered graphically by horizontal and vertical juxtaposition of boxes, respectively. (1) In any SMT there are specific Σ-terms generating the underlying symmetric monoidal structure: these are id 1 : 1 → 1, represented as , the symmetry σ 1,1 : 1 + 1 → 1 + 1, represented as , and the unit object for ⊕, that is, id 0 : 0 → 0, whose representation is an empty diagram . Graphical representation for arbitrary identities id n and symmetries σ n,m are generated according to the pasting rules in (1). Remark 2.4 (Models of a PROP). The assertion that (Σ M , E M ) is the SMT of commutative monoids-and similarly for other SMTs appearing in our expositioncan be made precise using the notion of model (sometimes also called algebra) of a PROP. Given a strict symmetric monoidal category C, a model of a PROP T in C is a symmetric strict monoidal functor F : T → C. Then LinMod(T , C) is the category of models of T in C and natural transformations between them. Turning to commutative monoids, there is a category Monoid(C) whose objects are the commutative monoids in C, i.e., objects x ∈ C equipped with arrows x ⊕ x → x and I → x, satisfying the usual equations. Given any model F : Mn → C, it follows that F(1) is a commutative monoid in C: this yields a functor LinMod(Mn, C) → Monoid(C). Saying that (Σ M , E M ) is the SMT of commutative monoids means that this functor is an equivalence natural in C. We shall not focus on models as they are not central in our developments and refer the reader to [19,25] for more information.
Example 2.3 only shows PROPs freely generated from an algebraic specification. However, one can also define PROPs in a more direct manner, without relying on SMTs. Two basic examples will be useful for our exposition: the PROP F whose arrows n → m are functions from n to m, where n = {0, 1, . . . , n − 1}. -the PROP P whose arrows n → m exist only if n = m, in which case they are the permutations on n. Similarly, P provides a concrete description of the theory (∅, ∅) with empty signature and no equations. It is the initial object in the category PROP.

PROP composition
A basic operation on SMTs (Σ, E) and (Σ , E ) is to take their sum (Σ Σ , E E ). In PROP, the PROP generated by (Σ Σ , E E ) is the coproduct T + S of the PROP T generated by (Σ, E) and S, generated by (Σ , E ).
The sum T +S is the least interesting way of combining theories, because there are no equations that express compatibility between the algebraic structures in T and S. This is a standard pattern in algebra: e.g. a ring is given by a monoid and an abelian group, subject to equations that ensure that the former distributes over the latter. Similarly, the equations of bialgebras (Example 2.3) describe the interplay of a monoid and a comonoid. Ordinary functions, which can always be decomposed as a surjection followed by an injection, are another example.
In [25] Lack shows how these phenomena can be uniformly described as the operation of composing PROPs. The conceptual switch is to understand PROPs as monads, and their composition as a distributive law. These monads live in a certain bicategory [3], as in the classical work by Street [36] called the unit and the multiplication respectively) making the following diagrams-in which we suppress the associativity isomorphisms-commute.
An epimorphic monad morphism is called a monad quotient.
For B = Cat, the above definition yields the standard notion of monad as an endofunctor with a pair of natural transformations. Something interesting happens for the case of the bicategory B = Span(Set) whose objects are sets, 1cells are spans of functions (with composition defined by pullback) and 2-cells are span morphisms: monads in Span(Set) are precisely the small categories. Indeed, which yields a set Ob of objects, one Ar of arrows and domain/codomain maps Ar ⇒ Ob . The unit η : id → F is a span morphism associating an identity arrow to each object (below left). The multiplication µ : F ; F → F is a span morphism defining composition for any two arrows a By thinking of categories as monads, one can define the composition of categories with the same set of objects as monad composition by a distributive law in Span(Set). This phenomenon is studied in [31].
in B making the following diagrams-in which we again omit associativity-commute.
A distributive law λ : F ; G → G ; F yields a monad G ; F with the following unit and multiplication: Let us verify how the abstract definition works for the case of categories. Pick categories C and D with the same set Ob of objects, seen as monads Ob As described in (7), λ allows to define a monad structure on D ; C. That means, λ yields a category D ; C whose arrows a → b are composable pairs a PROPs are understood as monads in the same sense that small categories are. The difference is that one needs to refine the bicategory of interest, in order for the composition of PROPs-monads to yield another PROP-monad and not an arbitrary small category. These refinements are in two steps. First, one takes the bicategory Span(Mon) whose 1-cells are spans in the category of monoids, instead of sets. This accounts for the monoidal structure -in fact, monads in Span(Mon) are small strict monoidal categories. The second refinement has 8 the purpose of correctly account for the symmetry structure of PROPs: one takes the bicategory Bimod(Span(Mon)) whose objects are small strict monoidal categories and 1-cells are the bimodules in Span(Mon). PROPs are then monads on the object P of Bimod(Span(Mon)).
We shall gloss over further details about the exact formalisation of this observation, as it is out of the scope of this paper -we refer to [25] and [37, § 2.4] for the detailed definitions. The simpler setting of composition of mere categories should provide enough guidance to follow the rest of our exposition.
It is important for our purposes to remark how composition works for PROPs T 1 , T 2 generated by SMTs, say (Σ 1 , E 1 ) and (Σ 2 , E 2 ). The PROP T 1 ; T 2 induced by a distributive law λ : T 2 ; T 1 → T 1 ; T 2 will also enjoy a presentation by generators and equations, consisting of the sum (Σ 1 Σ 2 , E 1 E 2 ) plus the equations E λ arising from the the distributive law. The set E λ is simply the graph of λ, now seen as a set of directed equations ( In fortunate cases, it is possible to present E λ by a simpler, or even finite, set of equations, thus giving a sensible axiomatisation of the compatibility conditions expressed by λ. This is the case for both examples considered below.  where the second diagram is obtained from the pullback by applying the isomorphisms F ∼ = Mn and F op ∼ = Cm. In fact, Lack [25] shows that the equations presenting Cm ; Mn arise from (those of Cm+Mn and) just four pullback squares, yielding equations (A7)-(A10). Therefore, Cm ; Mn is isomorphic to the PROP B of bialgebras encountered in Example 2.3. Furthermore, these PROPs have a "concrete" descriptions as F op ; F. In the terminology of [3], one can see F op ; F as the classifying category of the bicategory Span(F), obtained by identifying the isomorphic 1-cells and forgetting the 2-cells.
There is a tight relationship between distributive laws and factorisation systems. Distributive laws of small categories are in 1-1 correspondence with socalled strict factorisation systems [31], in which factorisations must be specified uniquely on the nose, rather than merely up-to isomorphism. Distributive laws of PROPs correspond instead to a more liberal kind of factorisation system, for which decompositions are up-to permutation. As this perspective will be useful later, we recall the following result from Lack [25].

Quotient of a distributive law
Definition 3.1 introduced the notion of quotient θ : F → G of a monad F: the idea is that the monad G is obtained by imposing additional equations on the algebraic theory described by F. As one may expect, distributive laws are compatible with monad quotients, provided that the law preserves the newly added equations. This folklore result appears in various forms in the literature: [9] gives it for distributive laws of endofunctors over monads and [4,8] for distributive laws of monads. All these references concern distributive laws in Cat. For our purposes, it is useful to state the result for arbitrary bicategories.
Then λ is a distributive law of monads.
Proof. The diagrams for compatibility of λ with unit and multiplication of G commute because θ is a monad morphism and (8) commutes. For compatibility of λ with unit and multiplication of H, one needs to use commutativity of (8) and the fact that θ is epi.
We remark that Proposition 3.5 holds also in the version in which one quotients the monad H instead of F. It is now useful to instantiate the result to the case of distributive laws of PROPs.
. Now, λ, λ and θ satisfy the assumptions of Proposition 3.5. In particular, (8) commutes by definition of λ in terms of λ and θ. The conclusion of Proposition 3.5 guarantees that λ is a distributive law. By construction, λ is presented by the same equations as λ.
We will see an application of Proposition 3.6 in the next section (Lemma 4.8).

Lawvere theories as composed PROPs
This section introduces and characterises Lawvere categories via a certain class of distributive laws of PROPs. As mentioned in the introduction, Lawvere categories are closely related to PROPs: the essential difference is that, whereas a Lawvere category is required to be a category with finite products -henceforth called a cartesian category, a PROP may carry any symmetric monoidal structure, not necessarily cartesian.
Just as PROPs, Lawvere categories can be also freely obtained by generators and equations. By analogy with symmetric monoidal theories introduced in § 2, we organise this data as a cartesian theory: it simply amounts to the notion of equational theory that one typically finds in universal algebra, see e.g. [13]. We use notation ovar (t) for the list of occurrences of variables appearing (from left to right) in t and, more generally, ovar (t 1 , . . . , t m ) for the list ovar (t 1 ) :: · · · :: ovar (t m ). Also, |l| ∈ N denotes the length of a list l. We say that a list t 1 , . . . , t m : n → m is linear if each variable among x 1 , . . . , x n appears exactly once in ovar (t 1 , . . . , t m ).
Our first observation is that Lawvere categories are PROPs.
Proof. Let × act as the monoidal product, 0 as its unit and define the symmetry n + m → m + n as the list x n+1 , . . . , x n+m , x 1 , . . . , x n . It follows that L (Σ,E) equipped with this structure satisfies the laws of symmetric strict monoidal categories, thus it is a PROP.
As a side observation, note that the unique PROP morphism P → L (Σ,E) given by initiality of P in PROP sends p : n → n to x p −1 (1) , . . . , x p −1 (n) .

Remark 4.3.
In spite of Proposition 4.2, cartesian theories are not a subclass of symmetric monoidal theories: in fact, the two concepts are incomparable. On the one hand, a symmetric monoidal theory (Σ, E) is cartesian if and only if all generators in Σ have coarity 1 and, for all equations t = s in E, t and s are Σ-terms with coarity 1. Under these conditions, there is a canonical way to interpret any Σ-term n → m as a list of m cartesian Σ-terms on variables x 1 , . . . , x n . Below, o ranges over Σ: The inductive cases are defined using the operations ; and ⊕ on lists given in Definition 4.1. Note that Σ-terms always denote (lists of) linear cartesian terms. This explains why, conversely, not all the cartesian theories are symmetric monoidal: their equations possibly involve non-linear Σ-terms, which are not expressible with (symmetric monoidal) Σ-terms. The subtlety here is that, in a sense, we can still simulate a cartesian theory on signature Σ with a symmetric monoidal theory, which however will be based on a larger signature Σ , recovering the possibility of copying and discarding variables by the use of additional generators. This point will become more clear below, where we will see how copier and discharger, i.e., the cartesian structure, can be mimicked with the use of the PROP Cm. The example of commutative monoids is particularly instructive for sketching our approach to Lawvere categories as composed PROPs. First, note that the Lawvere category L Σ M ,E M includes the PROP Mn freely generated by (Σ M , E M ). Indeed, any string diagram of Mn can be interpreted as a list of terms following the recipe of Remark 4.3. For instance, As we observed above, string diagrams of Mn can only express linear terms. What makes L Σ M ,E M more general than Mn is the ability to copy and discard variables. Indeed, just as any monoidal category in which ⊕ is the cartesian product, L Σ M ,E M comes equipped with canonical choices of a "copy" and "discard" operation cpy(n) := x 1 , . . . , x n , x 1 , . . . , x n : n → 2n dsc(n) := : n → 0 n ∈ N natural in n, which satisfy some expected equations: copying is commutative and associative; copying and then discarding is the same as not doing anything -see e.g. [18,14].
How can we mimic the copy and discard structure in the language of PROPs? First, for each n > 1 one can define cpy(n) and dsc(n) in terms of cpy(1) and dsc(1), which can therefore be regarded as the only fundamental operations 7 . Also, the equations that they satisfy can be synthesised as saying that cpy(1) acts as the comultiplication and dsc(1) as the counit of a commutative comonoid on 1. Therefore, they are none other than the generators of the PROP Cm: Our approach suggests that a copy of Mn and of Cm "live" inside L Σ M ,E M . We claim that these two PROPs provide a complete description of L Σ M ,E M , that means, any arrow of L Σ M ,E M can be presented diagrammatically by using Mn and Cm. For instance, Observe that the diagram is of the factorised form Intuitively, Cm is deputed to model the interplay of variables -in this case, the fact that x 1 is copied and x 3 is deleted -and Mn describes the syntactic tree of the terms. Of course, to claim that this factorisation is always possible, we need additional equations to model composition of factorised diagrams. For instance: The second equality holds if we assume the equation (A8) of the SMT of bialgebras. Thus the example suggests that composition by substitution in L Σ M ,E M can be mimicked at the diagrammatic level by allowing the use of bialgebra equations, which as we know from Example 3.3.(b) present the composite PROP Cm ; Mn. Therefore, the conclusive conjecture of our analysis is that L Σ M ,E M must be isomorphic to Cm ; Mn and can be presented by equations (A1)-(A10). We now generalise and make formal the above approach. Our main result is the following. Before moving to the proof of Theorem 4.5, we show its significance by revisiting some well-known theories in terms of our result.  Interestingly, the result of this quotient is isomorphic to the PROP of integer matrices, see e.g. [37, §3.5] and its models in a symmetric monoidal category are the Hopf algebras [17], with playing the role of the antipode. (c) In [21] Fritz presents the category of finite sets and probabilistic maps using generators and equations. The resulting Lawvere category L Prob can be decomposed following the scheme of Theorem 4.5: there is a linear part (Σ P , E P ) of the theory -given by binary convex combinations Σ P = { : 2 → 1 | λ ∈ [0, 1]} and suitable associativity and commutativity laws in E P , a commutative comonoid structure, and the two interact according to (Lw1)-(Lw2). This interaction yields a composite PROP Cm ; T (Σ P ,E P ) which, quotiented by non-linear equations 0 = and = , yields L Prob . 8 The rest of the section is devoted to proving Theorem 4.5. First we observe that, by the following lemma, it actually suffices to check our statement for SMTs with no equations. This reduction has just the purpose of making computations in L (Σ,E) easier, by working with terms instead of equivalence classes. Proof. Let (Σ, E) be a cartesian SMT and T Σ , T (Σ,E) be the PROPs freely generated, respectively, by (Σ, ∅) and (Σ, E). By assumption, Theorem 4.5 holds for (Σ, ∅), yielding a distributive law λ : T Σ ; Cm → Cm ; T Σ . A routine check shows that λ preserves the equations of E, whence Proposition 3.6 gives a distributive law λ : T (Σ,E) ; Cm → Cm ; T (Σ,E) with the required properties.
In the sequel, let us abbreviate L Σ,∅ as L Σ . By virtue of Lemma 4.8, we shall prove Theorem 4.5 for L Σ and by letting T Σ be the PROP freely generated by (Σ, ∅). We shall obtain the distributive law T Σ ; Cm → Cm ; T Σ from the recipe  -Cm is the sub-PROP of L Σ whose arrows are lists of variables. The inclusion of Cm in L Σ is the morphism Φ : Cm → L Σ defined on generators of Cm by -T Σ is the sub-PROP of L Σ whose arrows are linear terms. The inclusion of Proof. First, it is immediate to verify that lists of variables are closed under composition, monoidal product and include all the symmetries of L Σ : therefore, they form a sub-PROP. The same holds for linear terms.
We now consider the first statement of the lemma. There is a 1-1 correspondence between arrows n f ∈L Σ − −−− → m that are lists of variables and functions m → n: the function for f maps k, for 1 ≤ k ≤ m, to the index l of the variable x l appearing in position k in f . This correspondence yields an isomorphism between the sub-PROP of L Σ whose arrows are lists of variables and F op . Composing this isomorphism with Cm ∼ = − → F op yields Φ as in the statement of the lemma. For the second statement, faithfulness is immediate by the fact that arrows of T Σ are Σ-terms modulo the laws of SMCs, with no additional equations. One can easily verify that Ψ : T Σ → L Σ identifies the linear terms in L Σ following the observations in Remark 4.3.
Henceforth, for the sake of readability we shall not distinguish between Cm and the isomorphic sub-PROP of L Σ identified by the image of Φ, and similarly for T Σ and Ψ . Lemma 4.9 allows us to use L Σ as an environment where Cm and T Σ interact. The following statement guarantees the soundness of the interaction described by (Lw1)-(Lw2). It is useful to observe that (Lw1)-(Lw2) allows us to copy and discard not only the generators but arbitrary string diagrams of T Σ .  We can now show the factorisation lemma. Therefore, we turn to a proof of ( ‡). We describe a procedure to transform the string diagram The result is a string diagram p ;d ; c , where p only contains components and -i.e., it is a string diagram of P -andd is a string diagram of T Σ where does not appear.