Example 11.5.0.83. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories, and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{E}}$. Assume that $U(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then the following conditions are equivalent:
The morphism $f$ is $U$-cartesian.
The morphism $f$ is $U$-cocartesian.
The morphism $f$ is an isomorphism in $\operatorname{\mathcal{E}}$.
In particular, every isomorphism in $\operatorname{\mathcal{E}}$ is both $U$-cartesian and $U$-cocartesian.