Remark 4.8.5.23. Stated more informally, Corollary 4.8.5.22 asserts that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $n$-full if it is surjective up to homotopy on $n$-morphisms (having fixed source and target). For an alternative formulation of this heuristic, see Proposition 4.8.5.30 below.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$