Kerodon

Remark 7.1.1.2. Stated more informally, a natural transformation $\alpha : \underline{Y} \rightarrow u$ exhibits $Y$ as a limit of $u$ if and only if postcomposition with $\alpha $ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( \underline{X}, u )$ for each object $X \in \operatorname{\mathcal{C}}$. Similarly, a natural transformation $\beta : u \rightarrow \underline{Y}$ exhibits $Y$ as a colimit of $u$ if and only if precomposition with $\beta $ induces a homotopy equivalence $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(K,\operatorname{\mathcal{C}})}( u, \underline{Z} )$ for each object $Z \in \operatorname{\mathcal{C}}$.