Kerodon

Remark 4.8.5.18 (Change of Source). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $n \geq 0$ be an integer. If $F$ is an equivalence of $\infty $-categories, then $G$ is $n$-full if and only if $G \circ F$ is $n$-full. The “only if” direction follows immediately from Remark 4.8.5.15. For the converse, suppose that $G \circ F$ is $n$-full, and let $F^{-1}: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a homotopy inverse to $F$. Then $F^{-1}$ is an equivalence of $\infty $-categories; in particular, it is $n$-full (Remark 4.8.5.11). Applying Remark 4.8.5.15, we conclude that the composition $G \circ F \circ F^{-1}$ is also $n$-full. This composition is isomorphic to $G$, so $G$ is $n$-full as well (Remark 4.8.5.17).