Remark 7.5.6.11 (03CF)

Remark 7.5.6.11. Constructions 7.5.6.8 and 7.5.3.3 are closely related. Let $\operatorname{\mathcal{C}}$ be a small category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets, and let $\mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes. Combining Corollary 5.3.2.24 with Proposition 5.3.3.24, we obtain canonical isomorphisms of Kan complexes

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}^{+} )_{\bullet } & = & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \operatorname{sTr}_{ \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}})/\operatorname{\mathcal{C}}} ) \\ & \simeq & \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{N}_{\bullet }^{\mathscr {G}}(\operatorname{\mathcal{C}}) ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}_{+}, \mathscr {G} )_{\bullet }. \end{eqnarray*}

More generally, if $\mathscr {G}$ is a diagram of $\infty $-categories, we can identify $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G}^{+} )_{\bullet }$ with the full subcategory of $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}_{+}, \mathscr {G} )_{\bullet }$ spanned by those natural transformations $\alpha : \mathscr {F}_{+} \rightarrow \mathscr {G}$ having the property that, for each object $C \in \operatorname{\mathcal{C}}$, the diagram

\[ \alpha _{C}: \mathscr {F}_{+}(C) = \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } ) \rightarrow \mathscr {G}(C) \]

carries horizontal edges of $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}|_{ \operatorname{\mathcal{C}}_{/C} } )$ to isomorphisms in the $\infty $-category $\mathscr {G}(C)$.