$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Variant 8.3.3.15. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be a locally $\lambda $-small $\infty $-category, and let $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda })$ be a covariant Yoneda embedding for $\operatorname{\mathcal{C}}$. Let $\kappa = \mathrm{ecf}(\lambda )$ be the exponential cofinality of $\lambda $, let $K$ be a $\kappa $-small simplicial set, and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the following conditions are equivalent:
- $(1)$
The morphism $\overline{f}$ is a limit diagram in $\operatorname{\mathcal{C}}$.
- $(2)$
The composition $h_{\bullet } \circ \overline{f}$ is a limit diagram in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda })$.
Proof.
Since $K$ is $\kappa $-small, the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$ admits $K$-indexed limits (Example 7.6.7.4). For each object $X \in \operatorname{\mathcal{C}}$, let $\operatorname{ev}_{X}: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}^{< \lambda }) \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ denote the functor given by evaluation at $X$. By virtue of Proposition 7.1.6.1, condition $(2)$ is equivalent to the requirement that for each object $X \in \operatorname{\mathcal{C}}$, the composition
\[ K^{\triangleleft } \xrightarrow { \overline{f} } \operatorname{\mathcal{C}}\xrightarrow { h_{\bullet } } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}},\operatorname{\mathcal{S}}^{< \lambda }) \xrightarrow { \operatorname{ev}_{X} } \operatorname{\mathcal{S}}^{< \lambda } \]
is a limit diagram in the $\infty $-category $\operatorname{\mathcal{S}}^{< \lambda }$. Since the composite functor $(\operatorname{ev}_{X} \circ h_{\bullet }): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ is corepresentable by $X$, the equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 7.4.5.16 (and Remark 7.4.5.18).
$\square$