Proposition 4.5.6.11. Let $\operatorname{\mathcal{C}}$ be a small category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be an isofibrant diagram of simplicial sets. For every functor $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ and every subfunctor $\mathscr {E}_0 \subseteq \mathscr {E}$, the restriction map
\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( \mathscr {E}, \mathscr {F} )_{\bullet } \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}_0, \mathscr {F} )_{\bullet } \]
is an isofibration of simplicial sets. If the inclusion $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence, then $\theta $ is a trival Kan fibration.
Proof.
Let $B$ be a simplicial set and let $A \subseteq B$ be a simplicial subset. We wish to show that every lifting problem
4.42
\begin{equation} \begin{gathered}\label{equation:isofibrant-mapping-space} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \mathscr {F})_{\bullet } \ar [d]^{\theta } \\ B \ar [r] \ar@ {-->}[ur] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}_0, \mathscr {F})_{\bullet } } \end{gathered} \end{equation}
admits a solution, provided that either the inclusion map $A \hookrightarrow B$ is a categorical equivalence or the inclusion $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence. Unwinding the definitions, we see that the diagram (4.42) determines a natural transformation
\[ \alpha _0: ( \underline{A} \times \mathscr {E} ) \coprod _{ ( \underline{A} \times \mathscr {E}_0 ) } ( \underline{B} \times \mathscr {E}_0) \rightarrow \mathscr {F}, \]
and that solutions to (4.42) can be identified with extensions of $\alpha _0$ to a natural transformation $\alpha : \underline{B} \times \mathscr {E} \rightarrow \mathscr {F}$. By virtue of our assumption that $\mathscr {F}$ is isofibrant, we are reduced to proving that the inclusion map
\[ ( \underline{A} \times \mathscr {E} ) \coprod _{ ( \underline{A} \times \mathscr {E}_0 ) } ( \underline{B} \times \mathscr {E}_0) \hookrightarrow \underline{B} \times \mathscr {E} \]
is a levelwise categorical equivalence, which follows from Corollary 4.5.4.15.
$\square$