Remark 4.2.3.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. The following conditions are equivalent:
The functor $U$ is an isomorphism of categories.
The functor $U$ is a left covering functor which induces a bijection $\operatorname{Ob}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Ob}(\operatorname{\mathcal{C}})$.
The functor $U$ is a right covering functor which induces a bijection $\operatorname{Ob}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Ob}(\operatorname{\mathcal{C}})$.