Remark 9.1.3.9. In condition $(3)$ of Proposition 9.1.3.8, let $f_{!}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{D}$ be the functor given by covariant transport along $f$. Then condition $(\ast )$ is equivalent to either of the following:
- $(\ast ')$
There exists an object $Y \in \operatorname{\mathcal{E}}_{D}$ and a natural transformation $\beta : f_{!} \circ e \rightarrow \underline{Y}$ (in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{E}}_{D})$).
- $(\ast '')$
The diagram $(f_{!} \circ e): K \rightarrow \operatorname{\mathcal{E}}_{D}$ admits an extension to $K^{\triangleright }$.
The equivalence $(\ast ) \Leftrightarrow (\ast ')$ follows from Remark 5.1.3.8 (applied to the cocartesian fibration $\operatorname{Fun}(K,\operatorname{\mathcal{E}}) \rightarrow \operatorname{Fun}(K,\operatorname{\mathcal{C}})$), and the equivalence $(\ast ') \Leftrightarrow (\ast '')$ from Theorem 4.6.4.17. In the case where $U$ is a left fibration, we can further reformulate conditions $(\ast '')$ as follows:
- $(\ast ''')$
The diagram $(f_{!} \circ e): K \rightarrow \operatorname{\mathcal{E}}_{D}$ is nullhomotopic.