Kerodon

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Proposition 7.3.4.7. Let $\delta : \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty$-categories, let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be an isofibration of $\infty$-categories, and suppose we are given a lifting problem

7.17
$$\begin{gathered}\label{equation:existence-Kan-fiberwise} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}\ar [r]^-{ F_0 } \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\ar@ {-->}[ur]^{F} \ar [r] & \operatorname{\mathcal{E}}. } \end{gathered}$$

The following conditions are equivalent:

$(1)$

The lifting problem (7.17) has a solution $F: \operatorname{\mathcal{K}}\star _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is $U$-left Kan extended from $\operatorname{\mathcal{K}}$.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the associated lifting problem

$\xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{K}}_{C} \ar [r] \ar [d] & \operatorname{\mathcal{D}}\ar [d]^{U} \\ \operatorname{\mathcal{K}}_{C}^{\triangleright } \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{E}}}$

has a solution $\operatorname{\mathcal{K}}_{C}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ which is a $U$-colimit diagram.