Kerodon

Example 5.3.2.6 (Corepresentable Functors). Let $\operatorname{\mathcal{C}}$ be a category and let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ be the functor corepresented by an object $C \in \operatorname{\mathcal{C}}$, given by $h^{C}(D) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$. Let us abuse notation by regarding $h^{C}$ as a functor from $\operatorname{\mathcal{C}}$ to the category of simplicial sets (by identifying each morphism set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ with the corresponding discrete simplicial set). Combining Examples 5.3.2.5 and 5.2.6.5, we obtain a canonical isomorphism of simplicial sets $ \underset { \longrightarrow }{\mathrm{holim}}( h^{C} ) \simeq \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/ } )$.