Kerodon

Remark 7.4.0.2. For any cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleleft }$, the associated covariant diffraction functor $\mathrm{Df}: \overline{\operatorname{\mathcal{E}}}_{ \bf 0} \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ automatically factors through the full subcategory $\operatorname{Fun}^{\operatorname{CCart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ (see Construction 7.4.1.3). Similarly, for any cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$, the covariant refraction functor $\mathrm{Rf}: \operatorname{\mathcal{E}}\rightarrow \overline{\operatorname{\mathcal{E}}}_{\bf 1}$ automatically carries $U$-cocartesian edges of $\operatorname{\mathcal{E}}$ to isomorphisms in the $\infty $-category $\overline{\operatorname{\mathcal{E}}}_{\bf 1}$ (Remark 7.4.3.5).