Definition 4.8.5.1 (Full Functors). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. We say that $F$ is full if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{D}}}( F(X), F(Y) ) \]
is surjective on connected components.