Kerodon

Example 5.2.6.5. Let $\operatorname{\mathcal{C}}$ be a category, let $X$ be an object of $\operatorname{\mathcal{C}}$, and let $h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ denote the functor corepresented by $X$ (given on objects by the formula $h^{X}(Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$). Then the category of elements $\int _{\operatorname{\mathcal{C}}} h^{X}$ can be identified with the coslice category $\operatorname{\mathcal{C}}_{X/}$ of Variant 4.3.1.4. Similarly, if $h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ is the functor represented by $X$ (given on objects by $h_{X}(Y) = \operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,X)$), then the category of elements $\int ^{\operatorname{\mathcal{C}}} h_{X}$ can be identified with the slice category $\operatorname{\mathcal{C}}_{/X}$.