Exercise 4.2.6.4. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Show that:
The morphism $f$ is a left covering map if and only if the evaluation map
\[ \operatorname{ev}_{0}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 0\} , X) \times _{ \operatorname{Fun}( \{ 0\} , S)} \operatorname{Fun}( \Delta ^1, S) \]is an isomorphism of simplicial sets
The morphism $f$ is a right covering map if and only if the evaluation map
\[ \operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 1\} , X) \times _{ \operatorname{Fun}( \{ 1\} , S)} \operatorname{Fun}( \Delta ^1, S) \]is an isomorphism of simplicial sets.
The morphism $f$ is a covering map if and only if both $\operatorname{ev}_{0}$ and $\operatorname{ev}_{1}$ are isomorphisms.