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8.1.4 The Cospan Construction

Let $\operatorname{\mathcal{C}}_0$ be a category which admits pushouts. In ยง2.2.1, we introduced a $2$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}}_0)$ having the same objects, where $1$-morphisms from $X$ to $Y$ in $\operatorname{Cospan}(\operatorname{\mathcal{C}}_0)$ are cospans from $X$ to $Y$: that is, diagrams $X \xrightarrow {f} B \xleftarrow {g} Y$ in the category $\operatorname{\mathcal{C}}_0$ (see Example 2.2.2.1). In this section, we introduce a generalization of this construction, which will allow us to replace the ordinary category $\operatorname{\mathcal{C}}_0$ by an $\infty $-category. More precisely, we will associate to every simplicial set $\operatorname{\mathcal{C}}$ a simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ of cospans in $\operatorname{\mathcal{C}}$ (Construction 8.1.4.1). In the special case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0)$ is the nerve of a category $\operatorname{\mathcal{C}}_0$ which admits pushouts, we show that $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ can be identified with the Duskin nerve of the $2$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}}_0)$ (Corollary 8.1.4.12).

Construction 8.1.4.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set. For every integer $n \geq 0$, we let $\operatorname{Cospan}_{n}(\operatorname{\mathcal{C}})$ denote the collection of morphisms $\operatorname{Tw}( \Delta ^ n ) \rightarrow \operatorname{\mathcal{C}}$ in the category of simplicial sets. The construction $[n] \mapsto \operatorname{Cospan}_{n}(\operatorname{\mathcal{C}})$ depends functorially on the set $[n] = \{ 0 < 1 < \cdots < n \} $ as an object of the category $\operatorname{{\bf \Delta }}^{\operatorname{op}}$, and can therefore be viewed as a simplicial set. We will denote this simplicial set by $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ and refer to it as the simplicial set of cospans in $\operatorname{\mathcal{C}}$.

Remark 8.1.4.2. Let $n \geq 0$ be an integer. Then the simplicial set $\operatorname{Tw}( \Delta ^{n} )$ can be identified with the nerve of the partially ordered set $Q = \{ (i,j) \in [n]^{\operatorname{op}} \times [n]: i \leq j \} $ (see Example 8.1.0.5). Consequently, if $\operatorname{\mathcal{C}}$ is an arbitrary simplicial set, then $n$-simplices of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ can be identified with morphisms $\operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{\mathcal{C}}$, which we depict informally as diagrams

\[ \xymatrix@R =40pt@C=20pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & \cdots & X_{n-1,n-1} \ar [dr] \ar [dl] & & X_{n,n} \ar [dl] \\ & \cdots \ar [dr] & & \cdots \ar [dr] \ar [dl] & & \cdots \ar [dl] & \\ & & X_{0,n-1} \ar [dr] & & X_{1,n} \ar [dl] & & \\ & & & X_{0,n}. & & & \\ } \]

Example 8.1.4.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then:

  • Vertices of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ can be identified with vertices of $\operatorname{\mathcal{C}}$.

  • Let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$. Then edges of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ joining $X$ to $Y$ can be identified with pairs $(f,g)$, where $f: X \rightarrow B$ and $g: Y \rightarrow B$ are edges of $\operatorname{\mathcal{C}}$ having the same target.

Remark 8.1.4.4 (Duality). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\sigma $ be an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, which we identify with a morphism of simplicial sets $\operatorname{Tw}( \Delta ^{n} ) \rightarrow \operatorname{\mathcal{C}}$. Composing with the automorphism

\[ \operatorname{Tw}( \Delta ^{n} ) \xrightarrow {\sim } \operatorname{Tw}( \Delta ^{n} ) \quad \quad (i,j) \mapsto (n-j, n-i), \]

we obtain a new $n$-simplex $\overline{\sigma }$ of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. The construction $\sigma \mapsto \overline{\sigma }$ determines an isomorphism of simplicial sets $\tau : \operatorname{Cospan}(\operatorname{\mathcal{C}}) \simeq \operatorname{Cospan}(\operatorname{\mathcal{C}})^{\operatorname{op}}$, which can be described concretely as follows:

  • For every vertex $X \in \operatorname{\mathcal{C}}$, the morphism $\tau $ carries $X$ (regarded as a vertex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$) to itself.

  • Let $X$ and $Y$ be vertices of $\operatorname{\mathcal{C}}$, and let $e: X \rightarrow Y$ be an edge of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$, given by a pair of edges $(f: X \rightarrow B, g: Y \rightarrow B)$ of $\operatorname{\mathcal{C}}$. Then $\tau (e): Y \rightarrow X$ is the edge of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ given by the pair $(g,f)$.

Note that $\tau $ is involutive: that is, the composition

\[ \operatorname{Cospan}(\operatorname{\mathcal{C}}) \xrightarrow { \tau } \operatorname{Cospan}(\operatorname{\mathcal{C}})^{\operatorname{op}} \xrightarrow { \tau ^{\operatorname{op}} } \operatorname{Cospan}(\operatorname{\mathcal{C}}) \]

is the identity automorphism of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.

Example 8.1.4.5. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then Notation 8.1.1.5 supplies a projection map $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$, carrying each vertex $(f: X \rightarrow Y)$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to the vertex $Y \in \operatorname{\mathcal{C}}$. Under the bijection supplied by Proposition 8.1.4.7, we can identify $\lambda _{+}$ with a morphism of simplicial sets $\rho _{+}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$, which we display informally as a diagram

\[ X_0 \xrightarrow {f_1} X_1 \xrightarrow {f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n, \]

then $\rho _{-}(\sigma )$ is an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ which can be depicted informally as a diagram

\[ \xymatrix@R =20pt@C=20pt{ X_0 \ar [dr]^-{f_1} & & X_1 \ar [dr]^{f_2} \ar [dl]^{\operatorname{id}} & & \cdots \ar [dl] \ar [dr] & & X_{n-1} \ar [dr]^{f_ n} \ar [dl]^{\operatorname{id}} & & X_ n \ar [dl]^{\operatorname{id}} \\ & X_1 \ar [dr]^{f_2} & & X_2 \ar [dl]^{\operatorname{id}} \ar [dr]^{f_3} & \cdots & X_{n-1} \ar [dl]^{\operatorname{id}} \ar [dr]^{f_ n} & & X_{n} \ar [dl]^{\operatorname{id}} & \\ & & \cdots \ar [dr]^{f_{n-1}} & & \cdots \ar [dl]^{\operatorname{id}} \ar [dr]^{f_ n} & & \cdots \ar [dl]^{\operatorname{id}} & & \\ & & & X_{n-1} \ar [dr]^{f_ n} & & X_{n} \ar [dl]^{\operatorname{id}} & & & \\ & & & & X_ n. & & & & } \]

Similarly, the projection map $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ determines a morphism of simplicial sets $\rho _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$. If $\tau $ is an $n$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}$, which we display informally as a diagram

\[ Y_0 \xleftarrow {g_1} Y_1 \xleftarrow {g_2} Y_2 \leftarrow \cdots \xleftarrow {g_ n} Y_ n, \]

then $\rho _{-}(\tau )$ is an $n$-simplex of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ which is depicted informally by the diagram

\[ \xymatrix@R =20pt@C=20pt{ Y_0 \ar [dr]^-{\operatorname{id}} & & Y_1 \ar [dr]^{\operatorname{id}} \ar [dl]^{g_1} & & \cdots \ar [dl] \ar [dr] & & Y_{n-1} \ar [dr]^{\operatorname{id}} \ar [dl]^{g_{n-1}} & & Y_ n \ar [dl]^{g_ n} \\ & Y_0 \ar [dr]^{\operatorname{id}} & & Y_1 \ar [dl]^{g_1} \ar [dr]^{\operatorname{id}} & \cdots & Y_{n-2} \ar [dl]^{g_{n-2}} \ar [dr]^{\operatorname{id}} & & Y_{n-1} \ar [dl]^{g_{n-1}} & \\ & & \cdots \ar [dr]^{\operatorname{id}} & & \cdots \ar [dl]^{g_1} \ar [dr]^{\operatorname{id}} & & \cdots \ar [dl]^{g_2} & & \\ & & & Y_{0} \ar [dr]^{\operatorname{id}} & & Y_{1} \ar [dl]^{g_1} & & & \\ & & & & Y_0. & & & & } \]

Note that both $\rho _{-}$ and $\rho _{+}$ are monomorphisms of simplicial sets.

Construction 8.1.4.6. Let $\operatorname{\mathcal{D}}$ be a simplicial set and let $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{D}}$ be an $n$-simplex of $\operatorname{\mathcal{D}}$. Invoking the functoriality of the twisted arrow construction, we obtain a map $\operatorname{Tw}( \Delta ^{n} ) \xrightarrow { \operatorname{Tw}(\sigma ) } \operatorname{Tw}(\operatorname{\mathcal{D}})$, which we can identify with an $n$-simplex $u( \sigma )$ of the simplicial set $\operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$. The construction $\sigma \mapsto u(\sigma )$ is compatible with face and degeneracy maps, and therefore determines a morphism of simplicial sets $u: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$ which we will refer to as the unit map.

Proposition 8.1.4.7. Let $\operatorname{\mathcal{D}}$ be a simplicial set and let $u: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$ be the unit map of Construction 8.1.4.6. For every simplicial set $\operatorname{\mathcal{C}}$, the composite map

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\operatorname{\mathcal{D}}), \operatorname{\mathcal{C}}) & \rightarrow & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{C}}) ), \operatorname{Cospan}(\operatorname{\mathcal{D}}) ) \\ & \xrightarrow { \circ u} & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{D}}, \operatorname{Cospan}(\operatorname{\mathcal{C}}) ) \end{eqnarray*}

is a bijection.

Proof. Let us regard the simplicial set $\operatorname{\mathcal{C}}$ as fixed. For every simplicial set $\operatorname{\mathcal{D}}$, the unit map $u$ of Construction 8.1.4.6 determines a function

\[ \theta _{\operatorname{\mathcal{D}}}: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{D}}, \operatorname{Cospan}(\operatorname{\mathcal{C}}) ). \]

Using Remark 8.1.1.4, we see that the construction $\operatorname{\mathcal{D}}\mapsto \theta _{\operatorname{\mathcal{D}}}$ carries colimits (in the category of simplicial sets) to limits (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set})$). Consequently, to show that $\theta _{\operatorname{\mathcal{D}}}$ is a bijection, we may assume without loss of generality that $\operatorname{\mathcal{D}}= \Delta ^ n$ is a standard simplex (see Corollary 1.1.8.17). In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. $\square$

Corollary 8.1.4.8. The twisted arrow functor

\[ \operatorname{Tw}: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}\quad \quad \operatorname{\mathcal{D}}\mapsto \operatorname{Tw}(\operatorname{\mathcal{D}}) \]

has a right adjoint, given on objects by the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Cospan}(\operatorname{\mathcal{C}})$.

We now study the relationship between Construction 8.1.4.1 with the classical cospan construction (Example 2.2.2.1).

Construction 8.1.4.9. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts, and let $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ denote the $2$-category of Example 2.2.2.1. Suppose we are given another category $\operatorname{\mathcal{D}}$ and a functor $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$. We define a strictly unitary lax functor $F^{+}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ as follows:

  • For each $X \in \operatorname{\mathcal{D}}$, we define $F^{+}(X) = F( \operatorname{id}_{X} )$; here we regard the identity morphism $\operatorname{id}_ X: X \rightarrow X$ as an object of the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$.

  • For each morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{D}}$, we define $F^{+}(f)$ to be the $1$-morphism of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ given by the cospan

    \[ F( \operatorname{id}_ X ) \xrightarrow { F(\operatorname{id}_ X, f) } F(f) \xleftarrow { F( f, \operatorname{id}_ Y) } F( \operatorname{id}_ Y ). \]

    Note that this determines the values of $F^{+}$ on $2$-morphisms, since every $2$-morphism in $\operatorname{\mathcal{D}}$ is an identity $2$-morphism.

  • For every pair of composable morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in $\operatorname{\mathcal{D}}$, the composition constraint $\mu _{g,f}: F^{+}(g) \circ F^{+}(f) \Rightarrow F^{+}(g \circ f)$ is the $2$-morphism of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ corresponding to the map $F(f) \amalg _{ F( \operatorname{id}_ Y) } F(g) \rightarrow F( g \circ f )$ classifying the commutative diagram

    \[ \xymatrix@R =50pt@C=50pt{ F( \operatorname{id}_ Y ) \ar [r]^-{ F(f,\operatorname{id}_ Y)} \ar [d]^{F(\operatorname{id}_ Y,g)} & F( f ) \ar [d]^{F(\operatorname{id}_ X,g)} \\ F( g ) \ar [r]^-{F(f,\operatorname{id}_ Z)} & F( g \circ f) } \]

    in the category $\operatorname{\mathcal{C}}$.

Example 8.1.4.10. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts and let $n$ be a nonnegative integer. Applying Construction 8.1.4.9 in the special case where $\operatorname{\mathcal{D}}= [n]$, we obtain a function

\[ \{ \textnormal{Functors $\operatorname{Tw}([n]) \rightarrow \operatorname{\mathcal{C}}$} \} \xrightarrow {\sim } \{ \textnormal{Strictly unitary lax functors $[n] \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$} \} . \]

Using Propositions 1.2.2.1 and 8.1.1.9, we can identify the left hand side with the collection of $n$-simplices of the simplicial set $\operatorname{Cospan}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$. This construction depends functorially on $n$, and therefore determines a morphism of simplicial sets from $\operatorname{Cospan}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$ to the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) )$.

We can now formulate our main result.

Theorem 8.1.4.11. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories, where $\operatorname{\mathcal{C}}$ admits pushouts. Then Construction 8.1.4.9 induces a bijection of sets

\[ \{ \textnormal{Functors $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$} \} \xrightarrow {\sim } \{ \textnormal{Strictly unitary lax functors $F^{+}: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$} \} . \]

Corollary 8.1.4.12. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. Then the comparison map of Example 8.1.4.10 determines an isomorphism of simplicial sets

\[ \operatorname{Cospan}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) ). \]

Exercise 8.1.4.13. Show that Corollary 8.1.4.12 implies Theorem 8.1.4.11. That is, to prove Theorem 8.1.4.11 in general, it suffices to treat the special case where $\operatorname{\mathcal{D}}$ is a category of the form $[n] = \{ 0 < 1 < \cdots < n\} $ for $n \geq 0$.

Remark 8.1.4.14. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. The construction of the $2$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ of Example 2.2.2.1 involves some auxiliary choices: if $X \leftarrow B \rightarrow Y$ and $Y \leftarrow C \rightarrow Z$ are cospans in $\operatorname{\mathcal{C}}$, then their composition (as $1$-morphisms of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$) is given by $X \leftarrow (B \amalg _{Y} C) \rightarrow Z$, where the pushout $B \amalg _{Y} C$ is only well-defined up to (canonical) isomorphism. Corollary 8.1.4.12 supplies a description of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) )$ which does not depend on these choices. This shows, in particular, that the $2$-category $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is well-defined up to (non-strict) isomorphism; see Example 2.2.6.13.

Example 8.1.4.15. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. Then $2$-simplices $\sigma $ of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Cospan}(\operatorname{\mathcal{C}}) )$ can be identified with commutative diagrams

\[ \xymatrix@R =50pt@C=50pt{ X_{0,0} \ar [dr] & & X_{1,1} \ar [dr] \ar [dl] & & X_{2,2} \ar [dl] \\ & X_{0,1} \ar [dr] & & X_{1,2} \ar [dl] & \\ & & X_{0,2} & & } \]

in the category $\operatorname{\mathcal{C}}$. It follows from Theorem 2.3.2.5 that the $2$-simplex $\sigma $ is thin (in the sense of Definition 2.3.2.3) if and only if the square appearing in the diagram is a pushout: that is, it induces an isomorphism $X_{0,1} \amalg _{ X_{1,1} } X_{1,2} \rightarrow X_{0,2}$ in the category $\operatorname{\mathcal{C}}$.

Proof of Theorem 8.1.4.11. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories, where $\operatorname{\mathcal{C}}$ admits pushouts, and let $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}(\operatorname{\mathcal{C}})$ be a strictly unitary lax functor of $2$-categories. For every morphism $f: X \rightarrow Y$ in the category $\operatorname{\mathcal{D}}$, we can identify $G(f)$ with a cospan from $G(X)$ to $G(Y)$ in the category $\operatorname{\mathcal{C}}$, given by a diagram we will denote by $G(X) \xrightarrow { b_{-}(f) } B(f) \leftarrow { b_{+}(f) } G(Y)$. Our assumption that $G$ is strictly unitary guarantees the following:

$(\ast )$

For each object $X \in \operatorname{\mathcal{D}}$, the object $B( \operatorname{id}_ X )$ is equal to $G(X)$, and the maps $b_{-}(\operatorname{id}_ X): G(X) \rightarrow B( \operatorname{id}_ X )$ and $b_{+}( \operatorname{id}_ X): G( X) \rightarrow B( \operatorname{id}_ X )$ are the identity morphisms from $G(X)$ to itself in the category $\operatorname{\mathcal{C}}$.

For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, the composition constraint $\mu _{g,f}$ for the lax functor $G$ can be identified with a morphism from the pushout $B(f) \amalg _{ G(Y) } B(g)$ to $B( g \circ f)$, or equivalently with a pair of morphisms

\[ p(g,f): B(f) \rightarrow B(g \circ f) \quad \quad q(g,f): B(g) \rightarrow B(g \circ f) \]

satisfying $p(g,f) \circ b_{+}(f) = q(g,f) \circ b_{-}(g)$. The axioms for a lax functor (Definition 2.2.4.5) then translate to the following additional conditions:

$(a)$

For every morphism $f: X \rightarrow Y$ in the category $\operatorname{\mathcal{D}}$, $p( \operatorname{id}_ Y, f)$ is the identity morphism from $B(f)$ to itself.

$(b)$

For every morphism $f: X \rightarrow Y$ in the category $\operatorname{\mathcal{D}}$, $q( f, \operatorname{id}_ X )$ is the identity morphism from $B(f)$ to itself.

$(c)$

For every composable triple of $1$-morphisms $W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z$ in the category $\operatorname{\mathcal{D}}$, we have

\[ p(h \circ g, f) = p( h, g \circ f) \circ p(g,f) \quad \quad q(h, g \circ f ) = q(h \circ g, f) \circ q(h,g) \]
\[ p( h, g \circ f) \circ q(g,f) = q(h \circ g, f) \circ p( h, g). \]

We wish to show that there exists a unique functor of ordinary categories $F: \operatorname{Tw}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{C}}$ such that $G = F^{+}$, where $F^{+}$ is the lax functor associated to $F$ by Construction 8.1.4.9. For this condition to be satisfied, the functor $F$ must satisfy the following conditions:

$(0)$

For each object $X \in \operatorname{\mathcal{D}}$, we have $F( \operatorname{id}_ X ) = G(X)$ (this guarantees that $G$ and $F^{+}$ coincide on objects).

$(1)$

For each morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{D}}$ (regarded as an object of $\operatorname{Tw}(\operatorname{\mathcal{D}})$), we have $F(f) = B(f)$, and the morphisms $b_{-}(f)$ and $b_{+}(f)$ are given by $F( \operatorname{id}_ X, f)$ and $F(f, \operatorname{id}_ Y)$, respectively (this guarantees that $G$ and $F^{+}$ coincide on $1$-morphisms, and therefore also on $2$-morphisms).

$(2)$

For every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, the morphisms $p(g,f): B(f) \rightarrow B(g \circ f)$ and $q(g,f): B(g) \rightarrow B(g \circ f)$ are given by $F(\operatorname{id}_ X,g): F(f) \rightarrow F(g \circ f)$ and $F(f, \operatorname{id}_ Z): F(g) \rightarrow F(g \circ f)$, respectively (this guarantees that the composition constraints on $G$ and $F^{+}$ coincide).

Note that the value of $F$ on each object of $\operatorname{Tw}(\operatorname{\mathcal{D}})$ is determined by condition $(1)$. Moreover, if $(u,v)$ is a morphism from $f: X \rightarrow Y$ to $f': X' \rightarrow Y'$ in the category $\operatorname{Tw}(\operatorname{\mathcal{D}})$, then condition $(2)$ guarantees that $F(u,v)$ must be equal to the composition

\[ F(f) = B(f) \xrightarrow { q(f,u) } B(f \circ u) \xrightarrow {p(v, f \circ u)} B( v \circ f \circ u) = B(f') = F(f'). \]

This proves the uniqueness of the functor $F$.

To prove existence, we define $F$ on objects $f$ of $\operatorname{Tw}(\operatorname{\mathcal{D}})^{\operatorname{op}}$ by the formula $F(f) = B(f)$, and on morphisms $(u,v): f \rightarrow f'$ by the formula $F(u,v) = p(v, f \circ u) \circ q(f,u)$. For any morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we can use $(a)$ and $(b)$ to compute

\[ F(\operatorname{id}_ X, \operatorname{id}_ Y) = p( \operatorname{id}_{X}, f) \circ q( f, \operatorname{id}_ Y) = \operatorname{id}_{B(f)} \circ \operatorname{id}_{ B(f)} = \operatorname{id}_{ B(f) }, \]

so that $F$ carries identity morphisms in $\operatorname{Tw}(\operatorname{\mathcal{D}})$ to identity morphisms in $\operatorname{\mathcal{C}}$. To complete the proof that $F$ is a functor, we note that for every pair of composable morphisms

\[ (f: X \rightarrow Y) \xrightarrow { (u,v)} (f': X' \rightarrow Y') \xrightarrow { (u',v')} (f'': X'' \rightarrow Y'') \]

in the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{D}})$, the identities given in $(c)$ allow us to compute

\begin{eqnarray*} F(u', v') \circ F(u,v) & = & p(v', f' \circ u') \circ q(f', u') \circ p( v, f \circ u) \circ q(f, u) \\ & = & p( v', v \circ f \circ u \circ u' ) \circ q( v \circ f \circ u, u') \circ p( v, f \circ u ) \circ q(f,u) \\ & = & p( v', v \circ f \circ u \circ u' ) \circ p( v, f \circ u \circ u' ) \circ q( f \circ u, u' ) \circ q(f, u ) \\ & = & p( v' \circ v, f \circ u \circ u' ) \circ q( f, u \circ u' ) \\ & = & F(u \circ u', v' \circ v). \end{eqnarray*}

We now complete the proof by showing that the functor $F$ satisfies conditions $(0)$, $(1)$, and $(2)$. Condition $(0)$ is an immediate consequence of $(\ast )$. To prove $(2)$, we note that for any pair of composable morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$ in $\operatorname{\mathcal{D}}$, identities $(a)$ and $(b)$ yield equalities

\[ F( \operatorname{id}_ X, g) = p( g, f) \circ q(f, \operatorname{id}_ X) = p(g,f) \quad \quad F(f, \operatorname{id}_ Z) = p( \operatorname{id}_ Z, g \circ f) \circ q( g, f) = q(g,f). \]

To prove $(1)$, we note that if $f: X \rightarrow Y$ is a morphism in $\operatorname{\mathcal{D}}$, then we have

\begin{eqnarray*} F(\operatorname{id}_ X, f) & = & p( f, \operatorname{id}_ X \circ \operatorname{id}_ X) \circ q( \operatorname{id}_ X, \operatorname{id}_ X ) \\ & = & p(f, \operatorname{id}_ X ) \circ \operatorname{id}_{ G(X) } \\ & = & p(f, \operatorname{id}_ X) \circ b_{+}( \operatorname{id}_ X ) \\ & = & q(f, \operatorname{id}_ X ) \circ b_{-}( f ) \\ & = & \operatorname{id}_{ B(f) } \circ b_{-}(f) \\ & = & b_{-}(f), \end{eqnarray*}

and a similar calculation yields $F(f, \operatorname{id}_ Y) = b_{+}(f)$. $\square$