Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.2.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing an object $C \in \operatorname{\mathcal{C}}$ and let $e: A \rightarrow B$ be a left cofinal morphism of simplicial sets. Suppose we are given a diagram $f: B \rightarrow \operatorname{\mathcal{C}}$ and a natural transformation $\alpha : \underline{C} \rightarrow f$, where $\underline{C} \in \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ denotes the constant diagram taking the value $C$. Then $\alpha $ exhibits $C$ as a limit of $f$ (in the sense of Definition 7.1.1.1 if and only if the induced natural transformation $\alpha |_{A}: \underline{C}|_{A} \rightarrow f|_{A}$ exhibits $C$ as a limit of the diagram $f|_{A}$.

Proof. By virtue of Remark 7.1.1.7, we are free to modify the natural transformation $\alpha $ by a homotopy and may therefore assume that it corresponds to a morphism of simplicial sets $\Delta ^0 \diamond B \rightarrow \operatorname{\mathcal{C}}$ which factors through the categorical equivalence $\Delta ^0 \diamond B \twoheadrightarrow \Delta ^0 \star B$ of Theorem 4.5.8.8. In this case, the desired result follows from Corollary 7.2.2.3 and Remark 7.1.2.6. $\square$