Proposition 5.3.2.21. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, let $\operatorname{\mathcal{E}}$ be a simplicial set, and define $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formula $\mathscr {G}( C) = \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}})$. Then composition with the natural transformation $u_{\mathscr {F}}$ of Construction 5.3.2.20 induces an isomorphism of simplicial sets
\[ \Phi _{\mathscr {F}}: \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G} )_{\bullet }. \]
Proof.
For every object $C \in \operatorname{\mathcal{C}}$, let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the functor corepresented by $C$ (given by $h^{C}(D) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$, regarded as a discrete simplicial set). For every simplicial set $K$, let $\underline{K}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the constant functor taking the value $K$. For every functor $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ we have a coequalizer diagram
\[ {\coprod }_{ C \rightarrow D} h^{D} \times \underline{ \mathscr {F}(C) } \rightrightarrows {\coprod }_{ C } h^{C} \times \underline{ \mathscr {F}(C) } \rightarrow \mathscr {F} \]
in the category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$. Note that, if we regard the simplicial set $\operatorname{\mathcal{E}}$ as fixed, then the construction $\mathscr {F} \mapsto \Phi _{\mathscr {F}}$ carries colimits in $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ to limits in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$. We can therefore assume without loss of generality that the functor $\mathscr {F}$ factors as a product $h^{C} \times \underline{K}$, for some object $C \in \operatorname{\mathcal{C}}$ and some simplicial set $K$.
Fix an integer $n \geq 0$; we wish to show that $\Phi _{\mathscr {F}}$ induces a bijection from $n$-simplices of $\operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}})$ to $n$-simplices of $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G} )_{\bullet }$. Replacing $\operatorname{\mathcal{E}}$ by the simplicial set $\operatorname{Fun}(K \times \Delta ^ n, \operatorname{\mathcal{E}})$, we are reduced to proving that Construction 5.3.2.20 induces a bijection
\[ \Phi _0: \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \underset { \longrightarrow }{\mathrm{holim}}( h^{C} ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})}( h^{C}, \mathscr {G} ). \]
Let $\mathscr {G}_0: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor given on objects by the formula
\[ \mathscr {G}_0(C) = \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^0, \mathscr {G}(C) ) = \operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}). \]
Under the identification of $ \underset { \longrightarrow }{\mathrm{holim}}( h^{C} ) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} )$ of Example 5.3.2.6, the function $\Phi _0$ corresponds to the bijection $\mathscr {G}_0(C) \simeq \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set}) }( h^{C}, \mathscr {G}_0 )$ supplied by Yoneda's lemma.
$\square$