Corollary 7.4.1.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a diagram, and suppose that the morphism space $M = \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}, \mathscr {F} )$ is essentially small. Then $M$ is a limit of $\mathscr {F}$ in the $\infty $-category $\operatorname{\mathcal{S}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $Y$ be a Kan complex and let $u: Y \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{S}}}( \Delta ^0, Y)$ be the homotopy equivalence of Remark 5.5.1.5. As in the proof of Proposition 7.4.1.1, we observe that $u$ exhibits the constant functor $\underline{Y}_{\operatorname{\mathcal{C}}}$ as a copower of $\underline{ \Delta ^0 }_{\operatorname{\mathcal{C}}}$ by $Y$, and therefore induces a homotopy equivalence
\[ T: \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})}( \underline{Y}_{\operatorname{\mathcal{C}}}, \mathscr {F} ) \rightarrow \operatorname{Fun}(Y, \operatorname{Hom}_{\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) }( \underline{\Delta ^0}_{\operatorname{\mathcal{C}}}, \mathscr {F} ) ) = \operatorname{Fun}(Y, M). \]
Setting $Y = M$, it follows that there exists a natural transformation $\beta : \underline{M}_{\operatorname{\mathcal{C}}} \rightarrow \mathscr {F}$ such that $T( \beta )$ is homotopic to the identity morphism $\operatorname{id}_{M}$. In particular, $T( \beta )$ is a homotopy equivalence, so that $\beta $ exhibits $M$ as a limit of $\mathscr {F}$ (Proposition 7.4.1.1). $\square$