Corollary 5.2.2.21. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:
- $(1)$
The morphism $U$ is a covering map (Definition 3.1.4.1).
- $(2)$
The morphism $U$ is a left covering map (Definition 4.2.3.8) and, for every edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, the covariant transport functor $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is a bijection.
- $(3)$
The morphism $U$ is a right covering map (Definition 4.2.3.8) and, for every edge $f: C \rightarrow D$ of the simplicial set $\operatorname{\mathcal{C}}$, the contravariant transport morphism $f^{\ast }: \operatorname{\mathcal{E}}_{D} \rightarrow \operatorname{\mathcal{E}}_{C}$ is a bijection.