Remark 7.4.1.5 (02TF)

Remark 7.4.1.5. Suppose we are given a pullback diagram of small simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [d]^{U} \ar [r] & \overline{\operatorname{\mathcal{E}}} \ar [d]^{ \overline{U} } \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleleft }. } \]

Then the covariant diffraction functor $\mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ {\bf 0} } \rightarrow \operatorname{Fun}_{/ \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ of Construction 7.4.1.3 is an equivalence of $\infty $-categories if and only if the covariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleleft } }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$ (this is a restatement of Theorem 7.4.1.1).