Definition 4.2.3.8. Let $f: X \rightarrow S$ be a morphism of simplicial sets. We say that $f$ is a left covering map if, for every pair of integers $0 \leq i < n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r] \ar@ {^{(}->}[d] & X \ar [d]^{f} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & S } \]
admits a unique solution. We say that $f$ is a right covering map if the analogous condition holds for $0 < i \leq n$.