Example 11.10.2.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, let $f: C \rightarrow D$ be a morphism in $\operatorname{\mathcal{C}}$, and let $Y \mapsto \widetilde{f}_{Y}$ be a cleavage of $U$ over $f$. If $f: C \rightarrow D$ is an isomorphism in $\operatorname{\mathcal{C}}$ and $Y$ is an object of $\operatorname{\mathcal{E}}_{D}$, then $\widetilde{f}_{Y}: f^{\ast }(Y) \rightarrow Y$ is an isomorphism in $\operatorname{\mathcal{E}}$ (Example ***). In particular, if $f = \operatorname{id}_{D}$ is the identity morphism from an object $D$ to itself, then the construction
determines an isomorphism from $f^{\ast }$ to the identity functor $\operatorname{id}_{\operatorname{\mathcal{E}}_{D} }$ in the category of endofunctors $\operatorname{Fun}( \operatorname{\mathcal{E}}_{D}, \operatorname{\mathcal{E}}_{D} )$. We will denote the inverse isomorphism by $\epsilon _{D}: \operatorname{id}_{ \operatorname{\mathcal{E}}_{D} } \rightarrow f^{\ast }$.