Remark 4.1.1.5. The collection of inner fibrations is closed under pullback. That is, given a pullback diagram of simplicial sets

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

where $q$ is an inner fibration, the morphism $q'$ is also an inner fibration. Conversely, if $q'$ is an inner fibration and $f$ is surjective, then $q$ is an inner fibration.