Kerodon

Remark 4.8.5.17 (Isomorphism Invariance). Let $F_0, F_1: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functors of $\infty $-categories which are isomorphic (as objects of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$). Then $F_0$ is $n$-full if and only if $F_1$ is $n$-full. To see this, let $\operatorname{Isom}(\operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}})$ spanned by the isomorphisms (Example 4.4.1.14), so that the evaluation functors $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Isom}(\operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$ are equivalences of $\infty $-categories (Corollary 4.4.5.10). The assumption that $F_0$ and $F_1$ are isomorphic guarantees that there exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{Isom}(\operatorname{\mathcal{D}})$ satisfying $F_0 = \operatorname{ev}_0 \circ F$ and $F_1 = \operatorname{ev}_1 \circ F$. Using Remark 4.8.5.16, we see that $F_0$ is $n$-full if and only if $F$ is $n$-full. Similarly, $F_1$ is $n$-full if and only if $F$ is $n$-full.