Remark 5.1.5.6. Let $q: X \rightarrow S$ be an inner fibration of simplicial sets. The following conditions are equivalent:
The morphism $q$ is a locally cartesian fibration.
For every pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^-{q'} \ar [r] & X \ar [d] \\ \Delta ^1 \ar [r] & S, } \]the morphism $q'$ is a locally cartesian fibration.
For every pullback diagram
\[ \xymatrix@R =50pt@C=50pt{ X' \ar [d]^-{q'} \ar [r] & X \ar [d] \\ \Delta ^1 \ar [r] & S, } \]the morphism $q'$ is a cartesian fibration.