Definition 4.8.5.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
We say that $F$ is $0$-full if it is essentially surjective: that is, every object of $\operatorname{\mathcal{D}}$ is isomorphic to $F(X)$, for some object $X \in \operatorname{\mathcal{C}}$ (Definition 4.6.2.12).
We say that $F$ is $1$-full if it is full: that is, for objects $X,Y \in \operatorname{\mathcal{C}}$ having images $\overline{X} = F(X)$ and $\overline{Y} = F(Y)$ in $\operatorname{\mathcal{D}}$, the map
\[ \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) ) \rightarrow \pi _0( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y} ) ) \]is surjective (Definition 4.8.5.1).
For $n \geq 2$, we say that $F$ is $n$-full if, for every morphism $u: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ having image $\overline{u}: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{D}}$, the induced map
\[ \pi _{m}( \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y), u) \rightarrow \pi _{m}( \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \overline{X}, \overline{Y}), \overline{u} ) \]is injective for $m = n-2$ and surjective for $m = n-1$.