Kerodon

Remark 7.7.1.9 (Change of Target). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$ be diagrams indexed by the same simplicial set $K$, and let $\gamma : \mathscr {F} \rightarrow \mathscr {G}$ be a natural transformation. Suppose we are given a functor of $\infty $-categories $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ which preserves pullback squares. Set $\mathscr {F}' = U \circ \mathscr {F}$ and $\mathscr {G}' = U \circ \mathscr {G}$, so that $\gamma $ induces a natural transformation $\gamma ': \mathscr {F}' \rightarrow \mathscr {G}'$ of $K$-indexed diagrams in the $\infty $-category $\operatorname{\mathcal{C}}'$. If $\gamma $ is cartesian, then $\gamma '$ is cartesian. The converse holds if the functor $U$ is conservative (see Proposition 7.1.4.15).