$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark In the statement of Theorem, we can replace the smallness assumption on $\operatorname{\mathcal{C}}$ by the weaker assumption that for every object $Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y)$ is essentially small. Note that this latter condition cannot be omitted: if $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ is corepresentable by $X$, then $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is homotopy equivalent to the small Kan complex $\mathscr {F}(Y)$.