Corollary 4.4.3.9. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is a Kan complex. The following conditions are equivalent:
- $(1)$
The morphism $q$ is a covering map.
- $(2)$
The morphism $q$ is a left covering map.
- $(3)$
The morphism $q$ is a right covering map.