Remark 11.5.0.69 (Base Change). Suppose we are given a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^-{U} \ar [r] & \operatorname{\mathcal{E}}' \ar [d]^-{U'} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}', } \]
where $U$ and $U'$ are inner fibrations. Let $f: C \rightarrow D$ be a morphism of $\operatorname{\mathcal{C}}$ having image $f': C' \rightarrow D'$ in $\operatorname{\mathcal{C}}'$. Then a functor $F: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ is given by covariant transport along $f$ if and only if it is given by covariant transport along $f'$, when regarded as a functor from $\operatorname{\mathcal{E}}'_{C'}$ to $\operatorname{\mathcal{E}}'_{D'}$.