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Remark (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, then $\operatorname{\mathcal{D}}^{\mathscr {F}}$ is isofibrant (in the sense of Definition 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