Kerodon

Remark 7.7.1.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and suppose we are given morphisms $\gamma : \mathscr {F} \rightarrow \mathscr {G}$ and $\gamma ': \mathscr {F}' \rightarrow \mathscr {G}'$ in the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ which are isomorphic when viewed as objects of $\operatorname{Fun}( \Delta ^1 \times K, \operatorname{\mathcal{C}})$. Then $\gamma $ is cartesian if and only if $\gamma '$ is cartesian (see Remark 7.6.2.10). In particular, the condition that a natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {G}$ is cartesian depends only on the homotopy class $[\gamma ]$.