Remark 7.4.4.11. 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 7.4.4.9 is an equivalence of $\infty $-categories if the restriction functor
\[ \operatorname{Fun}_{ / \operatorname{\mathcal{C}}^{\triangleleft } }^{\operatorname{CCart}}( \operatorname{\mathcal{C}}^{\triangleleft }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]
is an equivalence of $\infty $-categories.