Kerodon

Example 7.1.1.4. Let $\operatorname{\mathcal{C}}$ be an ordinary category, let $K$ be a simplicial set, and suppose we are given a diagram $u: K \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we can identify with a functor of ordinary categories $U: \mathrm{h} \mathit{K} \rightarrow \operatorname{\mathcal{C}}$ (see Proposition 1.4.5.7). If $Y$ is an object of $\operatorname{\mathcal{C}}$, then we can use Corollary 1.5.3.5 to identify natural transformations $\underline{Y} \rightarrow u$ (of diagrams in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$) with natural transformations $\underline{Y} \rightarrow U$ (of diagrams in the ordinary category $\operatorname{\mathcal{C}}$). Under this identification, a natural transformation $\underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ (in the $\infty $-categorical sense of Definition 7.1.1.1) if and only if it exhibits $Y$ as a limit of $U$ (in the classical sense of Definition 7.1.0.1).