Remark 5.3.4.6 (Change of $\operatorname{\mathcal{E}}$). Suppose we are given a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^{T} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{ U' } \\ & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), & } \]
where the vertical maps are cocartesian fibrations and $T$ is an equivalence of cocartesian fibrations over $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. Then a morphism $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \rightarrow \operatorname{\mathcal{E}}$ is a scaffold of the cocartesian fibration $U$ if and only if $T \circ \lambda $ is a scaffold of the cocartesian fibration $U'$.