Remark 9.2.4.6. Let $\operatorname{\mathcal{C}}$ be an essentially small $\infty $-category, and let us abuse notation by identifying $\operatorname{\mathcal{C}}$ with a full subcategory of $\operatorname{Ind}(\operatorname{\mathcal{C}})$. Stated more informally, Proposition 9.2.4.5 asserts that the functor
\[ \operatorname{Ind}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \quad \quad C \mapsto \operatorname{Hom}_{\operatorname{Ind}(\operatorname{\mathcal{C}})}( \bullet , C) \]
is fully faithful, and that its essential image is spanned by the flat functors from $\operatorname{\mathcal{C}}^{\operatorname{op}}$ to $\operatorname{\mathcal{S}}$. In particular, a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ is flat if and only if it is representable by an object of $\operatorname{Ind}(\operatorname{\mathcal{C}})$.