$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Warning In the situation of Definition, the natural transformation $\alpha _{C}$ appearing in condition $(\ast _ C)$ is defined as a composition of morphisms in the $\infty $-category $\operatorname{Fun}( K_{/C}, \operatorname{\mathcal{D}})$, which is only well-defined up to homotopy. However, the condition that $\beta _{C}$ exhibits $F(C)$ as a colimit of the diagram $F_0|_{ K_{/C} }$ depends only on the homotopy class $[\beta _ C]$ (Remark