Remark 5.4.2.4. Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^-{q'} & \operatorname{\mathcal{C}}\ar [d]^-{q} \\ \operatorname{\mathcal{D}}' \ar [r]^-{F} & \operatorname{\mathcal{D}}. } \]
Assume that $\operatorname{\mathcal{D}}$ and $\operatorname{\mathcal{D}}'$ are $(\infty ,2)$-categories and that the morphism $F$ carries thin $2$-simplices of $\operatorname{\mathcal{D}}'$ to thin $2$-simplices of $\operatorname{\mathcal{D}}$ (that is, that $F$ is a functor of $(\infty ,2)$-categories; see Definition 5.4.7.1). If $q$ is an interior fibration, then $q'$ is an interior fibration.