Proposition 4.6.9.19. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, let $\theta _{X,Y}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}} }(X,Y)$ denote the homotopy equivalence of Kan complexes supplied by Remark 4.6.8.6. Then, for every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [d]^{ [\theta _{Y,Z} \times \theta _{X,Y}] }_{\sim } \ar [r]^-{\circ } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [d]^{ [ \theta _{X,Z} ]}_{\sim } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( Y,Z) \times \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X,Y) \ar [r]^-{\circ } & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X,Z) } \]
commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$; here the lower horizontal map is the composition law of Construction 4.6.9.9.
Proof.
We will show that there exists a morphism of Kan complexes
\[ \theta _{X,Y,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}(X,Y,Z) \]
for which the diagram
\[ \xymatrix@R =50pt@C=20pt{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \ar [d]^{\theta _{Y,Z} \times \theta _{X,Y}} \ar [rr]^-{\circ } \ar [dr]^{\theta _{X,Y,Z}} & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [d]^{ \theta _{X,Z} } \\ \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( Y,Z) \times \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X,Y) & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Y,Z) \ar [l] \ar [r] & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}( X,Z) } \]
is commutative.
Fix an integer $n \geq 0$. Let $\operatorname{\mathcal{E}}$ denote the simplicial category with object set $\operatorname{Ob}(\operatorname{\mathcal{E}}) = \{ x,y,z \} $ and morphism spaces given by
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( x,x )_{\bullet } = \{ \operatorname{id}_ x \} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}( y,y )_{\bullet } = \{ \operatorname{id}_ y \} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}(z,z)_{\bullet } = \{ \operatorname{id}_ z \} \]
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( y,x )_{\bullet } = \emptyset \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}(z,x)_{\bullet } = \emptyset \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}( z,y)_{\bullet } = \emptyset \]
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } = \Delta ^ n \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}(y,z)_{\bullet } = \Delta ^ n, \]
where the composition law $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(y,z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$ is an isomorphism (so that $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$ can be identified with the product $\Delta ^ n \times \Delta ^ n$). Note that there is a unique simplicial functor $F: \operatorname{Path}[ \Delta ^2 \times \Delta ^ n ]_{\bullet } \rightarrow \operatorname{\mathcal{E}}$ satisfying the following conditions:
On objects, the functor $F$ is given by the formula
\[ F(i,j) = \begin{cases} x & \text{ if $i=0$ } \\ y & \text{ if $i=1$} \\ z & \text{ if $i=2$.} \end{cases} \]
Let $(i,j)$ and $(i',j')$ be vertices of $\Delta ^2 \times \Delta ^ n$ satisfying $i < i'$ and $j \leq j'$, so that there is a unique indecomposable morphism $u$ from $(i,j)$ to $(i', j')$ in the path category $\operatorname{Path}[ \Delta ^2 \times \Delta ^ n ]$ (given by the chain $\{ (i,j) < (i',j') \} $). If $i=0$ and $i' = 1$, then $F(u)$ is the vertex $j'$ of $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet }$. If $i = 1$ and $i' =2$, then $F(u)$ is the vertex $j'$ of $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(y,z)_{\bullet }$. If $i= 0$ and $i' = 2$, then $F(u)$ is the vertex $(j',j')$ of $\Delta ^ n \times \Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,z)_{\bullet }$.
Let $\sigma $ and $\tau $ be $n$-simplices of the Kan complexes $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y,Z)_{\bullet }$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$, respectively. Then there is a unique simplicial functor $G_{\sigma ,\tau }: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ satisfying the following conditions:
On objects, the functor $G_{\sigma ,\tau }$ is given by $G_{\sigma ,\tau }(x) = X$, $G_{\sigma ,\tau }(y) = Y$, and $G_{\sigma ,\tau }(z) = Z$.
The induced map $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is the $n$-simplex $\tau $.
The induced map $\Delta ^ n = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( y,z)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$ is the $n$-simplex $\sigma $.
The composite simplicial functor
\[ \operatorname{Path}[ \Delta ^2 \times \Delta ^ n ]_{\bullet } \xrightarrow { F } \operatorname{\mathcal{E}}\xrightarrow { G_{\sigma ,\tau } } \operatorname{\mathcal{C}} \]
determines a morphism from $\Delta ^2 \times \Delta ^ n$ to the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which can be identified with an $n$-simplex $\theta _{X,Y,Z}(\sigma ,\tau )$ of the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y,Z)_{\bullet }$. Allowing $n$ to vary, the construction $(\sigma ,\tau ) \mapsto \theta _{X,Y,Z}(\sigma ,\tau )$ determines a morphism of simplicial sets $\theta _{X,Y,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})}(X,Y,Z)$ having the desired properties.
$\square$