$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 4.2.3.19. Let $f: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:

- $(1)$
The morphism $f$ is an isomorphism.

- $(2)$
The morphism $f$ is a left covering map and induces a bijection from the set of vertices of $X$ to the set of vertices of $S$.

- $(3)$
The morphism $f$ is a right covering map and induces a bijection from the set of vertices of $X$ to the set of vertices of $S$.

**Proof.**
The implications $(1) \Rightarrow (2)$ and $(1) \Rightarrow (3)$ are immediate. We will show that $(2) \Rightarrow (1)$; the proof that $(3) \Rightarrow (1)$ is similar. Assume that $f$ is a left covering map which is bijective at the level of vertices; we wish to show that every $n$-simplex $\sigma : \Delta ^ n \rightarrow S$ can be lifted uniquely to an $n$-simplex of $X$. Replacing $f$ by the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$, we may assume that $S = \Delta ^ n$ is a standard simplex (Remark 4.2.3.15). In this case, Proposition 4.2.3.16 guarantees that we can identify $f$ with the nerve of a left covering map of categories $F: \operatorname{\mathcal{E}}\rightarrow [n]$, so the desired result follows from Remark 4.2.3.5.
$\square$