Warning 7.3.1.4. In the situation of Definition 7.3.1.2, 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 $\alpha _{C}$ exhibits $F(C)$ as a colimit of the diagram $F_0|_{ K_{/C} }$ depends only on the homotopy class $[\beta _ C]$ (Remark 7.1.1.7).
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