Proposition 8.3.6.7. Let $\operatorname{\mathcal{C}}$ be a locally Kan simplicial category. Then the diagram (8.56) is a categorical pullback square.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Note that the vertical maps in the diagram (8.56) are left fibrations (Propositions 8.1.1.11 and 5.5.3.2). It will therefore suffice to show that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map of fibers
\begin{eqnarray*} \widetilde{\mathscr {H}}_{X,Y}: \{ X\} \times _{ \operatorname{\mathcal{E}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{E}}) \times _{\operatorname{\mathcal{E}}} \{ Y\} & \rightarrow & \{ \mathscr {H}_{\operatorname{\mathcal{C}}}(X,Y) \} \times _{ \operatorname{\mathcal{S}}} \operatorname{\mathcal{S}}_{\ast } \\ & = & \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{S}}}( \Delta ^0, \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet } ) \end{eqnarray*}
is a homotopy equivalence of Kan complexes (see Corollary 5.1.7.16). Note that the coslice inclusion of Construction 8.1.2.7 induces a monomorphism of simplicial sets $\iota : \operatorname{Hom}_{\operatorname{\mathcal{E}}}^{\mathrm{L}}(X,Y) \hookrightarrow \{ X\} \times _{ \operatorname{\mathcal{E}}^{\operatorname{op}} } \operatorname{Tw}(\operatorname{\mathcal{E}}) \times _{\operatorname{\mathcal{E}}} \{ Y\} $. Unwinding the definitions, we see that the composite map
\[ ( \widetilde{\mathscr {H}}_{X,Y} \circ \iota ): \operatorname{Hom}_{\operatorname{\mathcal{E}}}^{\mathrm{L}}(X,Y) \rightarrow \operatorname{Hom}^{\mathrm{L}}_{\operatorname{\mathcal{S}}}( \Delta ^0, \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X,Y)_{\bullet } ) \]
coincides with isomorphism described in Remark 4.6.8.18. It will therefore suffice to show that $\iota $ is a homotopy equivalence, which is a special case of Corollary 8.1.2.10. $\square$