Definition 8.4.1.1. Let $\operatorname{\mathcal{D}}$ be a (locally small) category. We say that a full subcategory $\operatorname{\mathcal{C}}\subseteq \operatorname{\mathcal{D}}$ is dense if the functor
\[ \operatorname{\mathcal{D}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{Set}) \quad \quad Y \mapsto \operatorname{Hom}_{\operatorname{\mathcal{D}}}( \bullet , Y ) \]
is fully faithful.