Remark 4.2.3.15. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d]^-{f'} & X \ar [d]^-{f} \\ S' \ar [r] & S. } \]

If $f$ is a left covering map, then $f'$ is a left covering map. If $f$ is a right covering map then $f'$ is a right covering map.

Conversely, suppose that $f: X \rightarrow S$ is a morphism of simplicial sets having the property that, for every $n$-simplex $\Delta ^ n \rightarrow S$, the projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ is a left covering map. Then $f$ is left covering map. If every projection map $\Delta ^ n \times _{S} X \rightarrow \Delta ^ n$ is a right covering map, then $f$ is a right covering map.