Kerodon

Remark 7.7.1.10 (Change of Source). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\mathscr {F}, \mathscr {G}: K \rightarrow \operatorname{\mathcal{C}}$ be diagrams indexed by the same simplicial set $K$, and let $\gamma : F \rightarrow G$ be a natural transformation. Suppose we are given a morphism of simplicial sets $u: K' \rightarrow K$. Set $\mathscr {F}' = \mathscr {F} \circ u$ and $\mathscr {G}' = \mathscr {G} \circ u$, so that $\gamma $ induces a natural transformation $\gamma ': \mathscr {F}' \rightarrow \mathscr {G}'$ of $K'$-indexed diagrams in $\operatorname{\mathcal{C}}$. If $\gamma $ is cartesian, then $\gamma '$ is cartesian. The converse holds if $\gamma $ is surjective on edges.