# Kerodon

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### 8.1.3 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 8.1.3.1). 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 8.1.3.15).

Construction 8.1.3.1. 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 8.1.3.2. 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 8.1.3.3. 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 8.1.3.4 (Symmetry). 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 $\operatorname{Cospan}(\operatorname{\mathcal{C}})$.

Construction 8.1.3.5. 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 operators, 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.

Example 8.1.3.6 (Tautological Cospans). Let $\operatorname{\mathcal{D}}$ be a simplicial set and let $e: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{D}}$, which we also view as a vertex of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{D}})$. Then the morphism $\operatorname{\mathcal{D}}\rightarrow \operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$ carries $e$ to an edge of the simplicial set $\operatorname{Cospan}( \operatorname{Tw}(\operatorname{\mathcal{D}}) )$, which we can identify with a pair of edges

$\operatorname{id}_{X} \xrightarrow { e_{L} } e \xleftarrow {e_{R}} \operatorname{id}_{Y}$

in the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{D}})$. Here $e_{L}$ and $e_{R}$ can be identified with degenerate $3$-simplices of $\operatorname{\mathcal{D}}$, which we depict informally in the diagrams

$\xymatrix@R =50pt@C=50pt{ X \ar [d]^{\operatorname{id}_ X} & X \ar [l]^{\operatorname{id}_{X}} & Y \ar [d]^{\operatorname{id}_ Y} & X \ar [l]^{f} \\ X \ar [r]^-{f} & Y & Y \ar [r]^-{\operatorname{id}_{Y}} & Y. }$

Proposition 8.1.3.7. 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 8.1.3.5. 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{D}}) ), \operatorname{Cospan}(\operatorname{\mathcal{C}}) ) \\ & \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 8.1.3.5 determines a function

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

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 Remark 1.1.3.13). In this case, the desired result follows immediately from the definition of the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$. $\square$

Corollary 8.1.3.8. 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 $\operatorname{\mathcal{C}}\mapsto \operatorname{Cospan}(\operatorname{\mathcal{C}})$.

Remark 8.1.3.9. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be simplicial sets. Using Proposition 8.1.3.7, we obtain bijections

\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Cospan}(\operatorname{\mathcal{D}}) ) & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n \times \operatorname{\mathcal{C}}, \operatorname{Cospan}(\operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}( \Delta ^ n \times \operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\Delta ^ n) \times \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\Delta ^ n), \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, \operatorname{Cospan}( \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) ) ). \end{eqnarray*}

These bijections depend functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore determine an isomorphism of simplicial sets $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Cospan}(\operatorname{\mathcal{D}}) ) \simeq \operatorname{Cospan}( \operatorname{Fun}( \operatorname{Tw}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) )$.

Corollary 8.1.3.10. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Kan fibration of simplicial sets. Then the induced map $\operatorname{Cospan}(U): \operatorname{Cospan}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Cospan}(\operatorname{\mathcal{D}})$ is also a Kan fibration.

Proof. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets which is a weak homotopy equivalence. We wish to show that every lifting problem

8.5
$$\begin{gathered}\label{equation:cospan-Kan-fibration} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d]^{i} & \operatorname{Cospan}(\operatorname{\mathcal{C}}) \ar [d]^{ \operatorname{Cospan}(U) } \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{Cospan}(\operatorname{\mathcal{D}}) } \end{gathered}$$

admits a solution. Using Proposition 8.1.3.7, we can rewrite (8.5) as a lifting problem of the form

$\xymatrix@R =50pt@C=50pt{ \operatorname{Tw}(A) \ar [r] \ar [d]^{\operatorname{Tw}(i)} & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \operatorname{Tw}(B) \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}. }$

Our assumption that $U$ is a Kan fibration guarantees that this lifting problem has a solution, since the monomorphism $\operatorname{Tw}(i): \operatorname{Tw}(A) \hookrightarrow \operatorname{Tw}(B)$ is also a weak homotopy equivalence (Corollary 8.1.2.6). $\square$

Corollary 8.1.3.11. Let $\operatorname{\mathcal{C}}$ be a Kan complex. Then the simplicial set $\operatorname{Cospan}(\operatorname{\mathcal{C}})$ is also a Kan complex.

Proof. Apply Corollary 8.1.3.10 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$. $\square$

We now study the relationship between Construction 8.1.3.1 with the classical cospan construction (Example 2.2.2.1).

Construction 8.1.3.12. 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 8.1.3.13. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts and let $n$ be a nonnegative integer. Applying Construction 8.1.3.12 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 1.3.3.1 and 8.1.1.10, 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 8.1.3.14. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be categories, where $\operatorname{\mathcal{C}}$ admits pushouts. Then Construction 8.1.3.12 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 8.1.3.15. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. Then the comparison map of Example 8.1.3.13 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 8.1.3.16. Show that Corollary 8.1.3.15 implies Theorem 8.1.3.14. That is, to prove Theorem 8.1.3.14 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 8.1.3.17. 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 $X \rightarrow B \leftarrow Y$ and $Y \rightarrow C \leftarrow Z$ are cospans in $\operatorname{\mathcal{C}}$, then their composition (as $1$-morphisms of $\operatorname{Cospan}(\operatorname{\mathcal{C}})$) is given by $X \rightarrow (B \amalg _{Y} C) \leftarrow Z$, where the pushout $B \amalg _{Y} C$ is only well-defined up to (canonical) isomorphism. Corollary 8.1.3.15 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 8.1.3.18. 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 8.1.3.14. 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 8.1.3.12. 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 $F(f, \operatorname{id}_ Y) = b_{+}(f)$. $\square$