Remark 7.5.1.6. Let $\operatorname{\mathcal{C}}$ be a category. For each object $C \in \operatorname{\mathcal{C}}$, let $\mathscr {E}(C)$ denote the simplicial set $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{ / C} )$. The construction $C \mapsto \mathscr {E}(C)$ determines a functor $\mathscr {E}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$, which we view as an object of the functor category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$. For every diagram of Kan complexes $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$, Proposition 5.3.3.24 supplies a canonical isomorphism
where the right hand side is defined using the simplicial enrichment of $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }})$ described in Example 2.4.2.2.
Stated more concretely, we can identify $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F})$ with a simplicial subset of the product ${\prod }_{C \in \operatorname{\mathcal{C}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ), \mathscr {F}(C) )$, whose $n$-simplices are collections of maps $\{ \sigma _{C}: \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \rightarrow \mathscr {F}(C) \} $ which satisfy the following condition:
- $(\ast )$
For every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{n} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \ar [r]^-{\circ f} \ar [d]^{ \sigma _{C} } & \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/D} ) \ar [d]^{ \sigma _ D } \\ \mathscr {F}(C) \ar [r]^-{ \mathscr {F}(f) } & \mathscr {F}(D) } \]is commutative.
In particular, we have an equalizer diagram of simplicial sets