Kerodon

Example 7.3.7.8. In the situation of Corollary 7.3.7.6, suppose that $U$ is a cartesian fibration. Let $U_{ \overline{e} }: \Delta ^1 \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1$ denote the cartesian fibration given by projection onto the first factor. By virtue of Proposition 7.3.3.11, the functor $f$ is $V$-left Kan extended from $\{ 0\} \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$ if and only if it carries $U_{ \overline{e} }$-cartesian morphisms of $\Delta ^1 \times _{\operatorname{\mathcal{B}}} \operatorname{\mathcal{C}}$ to $V$-cocartesian morphisms of $\operatorname{\mathcal{D}}$. In this case, Corollary 7.3.7.6 is a special case of (the dual of) Lemma 5.3.6.11.