Definition 4.2.3.1. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a functor between categories. We say that $U$ is a left covering functor if it satisfies the following condition:
For every object $X \in \operatorname{\mathcal{E}}$ and every morphism $\overline{f}: U(X) \rightarrow \overline{Y}$ in the category $\operatorname{\mathcal{C}}$, there is a unique pair $(Y,f)$, where $Y$ is an object of $\operatorname{\mathcal{E}}$ with $U(Y) = \overline{Y}$ and $f: X \rightarrow Y$ is a morphism in $\operatorname{\mathcal{E}}$ with $U(f) = \overline{f}$.
We say that $U$ is a right covering functor if it satisfies the following dual condition:
For every object $Y \in \operatorname{\mathcal{E}}$ and every morphism $\overline{f}: \overline{X} \rightarrow U(Y)$ in the category $\operatorname{\mathcal{C}}$, there is a unique pair $(X,f)$, where $X$ is an object of $\operatorname{\mathcal{E}}$ satisfying $U( X ) = \overline{X}$ and $f: X \rightarrow Y$ is a morphism in $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.