Kerodon

Remark 7.4.3.2. The proof of Proposition 7.4.3.1 given in this section will proceed by reduction to the special case where $\mathscr {F}$ is the constant functor $\underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}$, using the theory of Kan extensions developed in §7.3. In §7.4.5, we will formulate a more general result which describes the colimit of the covariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ for a cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ which is not assumed to be a left fibration (see Proposition 7.4.5.1 and Remark 7.4.5.4). For this, we give a different proof which is independent of the results of this section (and does not use the theory of Kan extensions).