Proof of Proposition 5.5.2.22.
Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category containing an object $X$, and let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{C}}_{X/}$ be a full simplicial subcategory satisfying hypothesis $(\ast )$ of Proposition 5.5.2.22. For every pair of objects $(Y,f), (Z,g) \in \operatorname{\mathcal{E}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet }$ is the fiber of the Kan fibration
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, Z)_{\bullet } \xrightarrow { \circ f} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \]
over the vertex $g$, and is therefore a Kan complex (Remark 3.1.1.9). Applying Theorem 2.4.5.1, we conclude that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})$ is an $\infty $-category. We wish to show that, for every pair of objects $(Y,f), (Z,g) \in \operatorname{\mathcal{E}}$ as above, the coslice comparison morphism $c'$ induces a homotopy equivalence of morphism spaces
\[ \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}) }( (Y,f), (Z,g) ) \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{X/ } }( (Y,f), (Z,g) ). \]
By virtue of Proposition 4.6.5.10, this is equivalent to the requirement that $c'$ induces a homotopy equivalence $\rho : \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}})}( (Y,f), (Z,g) ) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/} }( (Y,f), (Z,g) )$ of left-pinched morphism spaces.
Construction 4.6.8.3 supplies comparison maps
\[ \overline{\theta }: \operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{E}}) }^{\mathrm{L}}( (Y,f), (Z,g) ) \]
\[ \theta _{Y,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(Y,Z) \quad \quad \theta _{X,Z}: \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}(X,Z), \]
which are homotopy equivalences of Kan complexes by virtue of Theorem 4.6.8.5. Let us regard $f: X \rightarrow Y$ as an edge of the simplicial set $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, and let $Q$ denote the fiber $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z\} $. Since the inclusion $\{ 1\} \hookrightarrow \Delta ^1$ is right anodyne, the restriction map $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ Y/}$ is a trivial Kan fibration (Proposition 4.3.6.12), and therefore restricts to a trivial Kan fibration
\[ \pi : Q \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ Y/} \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z \} = \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})} (Y, Z). \]
In particular, $Q$ is a Kan complex and $\pi $ is a homotopy equivalence. Let $\pi '$ denote the restriction map
\[ Q \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ X/} \times _{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}) } \{ Z \} = \operatorname{Hom}^{\mathrm{L}}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})} (X, Z). \]
Note that $\pi '$ is a pullback of the left fibration $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ f/ } \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})_{ X/}$ (Corollary 4.3.6.11), and is therefore also a left fibration (Remark 4.2.1.8). Since the left-pinched morphism space $\operatorname{Hom}_{ \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z)$ is a Kan complex (Proposition 4.6.5.5), the morphism $\pi '$ is a Kan fibration (Corollary 4.4.3.8). We will construct an auxiliary map of Kan complexes $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$ with the following properties:
- $(a)$
The composition $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \xrightarrow { \lambda } Q \xrightarrow {\pi } \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(Y,Z)$ is equal to $\theta _{Y,Z}$.
- $(b)$
The cubical diagram of Kan complexes
5.55
\begin{equation} \begin{gathered}\label{equation:diagram-Kan-slice-compatibility} \xymatrix@C =-10pt@R=40pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( (Y,f), (Z,g) )_{\bullet } \ar [rr] \ar [dr]_{\rho \circ \overline{\theta } } \ar [dd] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \ar [dr]_{\lambda } \ar [dd]^(.6){\circ f} & \\ & \operatorname{Hom}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})_{X/}}^{\mathrm{L}}( (Y,f), (Z,g) ) \ar [rr] \ar [dd] & & Q \ar [dd]^{\pi '} \\ \{ g\} \ar@ {=}[dr] \ar [rr] & & \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \ar [dr]_-{\theta _{X,Z}} & \\ & \{ g\} \ar [rr] & & \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z) } \end{gathered} \end{equation}
is commutative.
Suppose that such a map has been constructed. It follows from $(a)$ that $\lambda $ is a homotopy equivalence. Moreover, the front and back faces of the diagram (5.55) are pullback squares of simplicial sets. Since the vertical maps
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet } \quad \quad \pi ': Q \rightarrow \operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}( X, Z) \]
are Kan fibrations, these faces are also homotopy pullback squares (Example 3.4.1.3). Since $\lambda $, $\theta _{X,Z}$, and the identity map $\operatorname{id}: \{ g\} \rightarrow \{ g\} $ are homotopy equivalences of Kan complexes, it follows from Corollary 3.4.1.12 that the map $\rho \circ \overline{\theta }$ is also a homotopy equivalence of Kan complexes. Since $\overline{\theta }$ is a homotopy equivalence, we conclude that $\rho $ is a homotopy equivalence as desired.
We now complete the proof by constructing the morphism $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$. Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet }$, so that $\theta _{Y,Z}( \sigma )$ is an $n$-simplex of the left-pinched morphism space $\operatorname{Hom}_{\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})}^{\mathrm{L}}(Y,Z)$ which we can identify with a simplicial functor $F_{\sigma }: \operatorname{Path}[ \{ y\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ such that $F_{\sigma }(y) = Y$ and $F_{\sigma } |_{ \operatorname{Path}[n]_{\bullet } }$ is the constant functor taking the value $Z$ (see Construction 4.6.8.3). We extend $F_{\sigma }$ to a simplicial functor $F_{\sigma }^{+}: \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ as follows:
The functor $F_{\sigma }^{+}$ carries $x$ to the object $X \in \operatorname{\mathcal{C}}$.
For every element $i \in \{ y\} \star [n]$, the induced map of simplicial sets
\[ \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ] }( x, i)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, F_{\sigma }(i) )_{\bullet } \]
is given by the composition
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Path}[ \{ x\} \star \{ y\} \star [n] ] }( x, i)_{\bullet } & \xrightarrow {u} & \operatorname{Hom}_{ \operatorname{Path}[ \{ y\} \star [n] ] }( y, i)_{\bullet } \\ & \xrightarrow { F_{\sigma } } & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( Y, F_{\sigma }(i) )_{\bullet } \\ & \xrightarrow { \circ f} & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, F_{\sigma }(i) )_{\bullet }, \end{eqnarray*}
where $u$ is induced by the map of partially ordered sets $\{ x\} \star \{ y\} \star [n] \rightarrow \{ y\} \star [n]$ which is the identity on $\{ y\} \star [n]$ and carries $x$ to $y$.
Then $F_{\sigma }^{+}$ determines a morphism of simplicial sets $\{ x \} \star \{ y\} \star \Delta ^{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ carrying $\{ x\} \star \{ y\} $ to the edge $f$ and $\Delta ^{n}$ to the vertex $Z$, which we can identify with an $n$-simplex $\lambda (\sigma )$ of the Kan complex $Q$. The construction $\sigma \mapsto \lambda (\sigma )$ depends functorially on $[n] \in \operatorname{{\bf \Delta }}$, and therefore induces a morphism of simplicial sets $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z)_{\bullet } \rightarrow Q$ which is easily verified to satisfy conditions $(a)$ and $(b)$.
$\square$