Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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