# Kerodon

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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 $\operatorname{Ob}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{Ob}(\operatorname{\mathcal{C}})$.