Definition 8.4.6.3. Let $\widehat{\operatorname{\mathcal{C}}}$ be an $\infty $-category. We say that a full subcategory $\operatorname{\mathcal{C}}\subseteq \widehat{\operatorname{\mathcal{C}}}$ is weakly dense if the following condition is satisfied:
Let $f: Y \rightarrow Z$ be a morphism of $\widehat{\operatorname{\mathcal{C}}}$ such that, for every object $X \in \operatorname{\mathcal{C}}$, the induced map
\[ \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}( X, Y) \xrightarrow { [f] \circ } \operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(X, Z) \]is a homotopy equivalence of Kan complexes. Then $f$ is an isomorphism.
We say that a collection of objects $\{ X_ i \} _{i \in I}$ of $\widehat{\operatorname{\mathcal{C}}}$ is weakly dense if it spans a weakly dense full subcategory of $\widehat{\operatorname{\mathcal{C}}}$.