# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 7.5.6.6 (Relationship to Isofibrant Diagrams). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $\operatorname{\mathcal{D}}$ be an $\infty$-category, and let $\operatorname{\mathcal{D}}^{\mathscr {F}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ denote the functor given by the construction $C \mapsto \operatorname{Fun}( \mathscr {F}(C), \operatorname{\mathcal{D}})$. If $\mathscr {F}$ is projectively cofibrant (in the sense of Definition 7.5.6.1), then $\operatorname{\mathcal{D}}^{\mathscr {F}}$ is isofibrant (in the sense of Definition 4.5.6.3). That is, if $\mathscr {E}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set_{\Delta }}$ is a diagram of simplicial sets and $\mathscr {E}_0 \subseteq \mathscr {E}$ is a subfunctor for which the equivalence $\mathscr {E}_0 \hookrightarrow \mathscr {E}$ is a levelwise categorical equivalence, then the restriction map

$\theta : \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}, \operatorname{\mathcal{D}}^{\mathscr {F}} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set_{\Delta }}) }( \mathscr {E}_0, \operatorname{\mathcal{D}}^{\mathscr {F}} )$

is surjective. This follows from the observation that $\theta$ can be identified with the map

$\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \operatorname{\mathcal{D}}^{\mathscr {E}} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \operatorname{\mathcal{D}}^{\mathscr {E}_0} )$

given by composition with the restriction map $\operatorname{\mathcal{D}}^{\mathscr {E}} \rightarrow \operatorname{\mathcal{D}}^{\mathscr {E}_0}$, which is a levelwise trivial Kan fibration by virtue of Corollary 4.5.5.19.