Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.1.7.16. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:

$(1)$

Let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $U \circ \overline{f}$ is a limit diagram in $\operatorname{\mathcal{D}}$. Then $\overline{f}$ is a limit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a $U$-limit diagram.

$(2)$

Let $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets such that $U \circ \overline{g}$ is a colimit diagram in $\operatorname{\mathcal{D}}$. Then $\overline{g}$ is a colimit diagram in $\operatorname{\mathcal{C}}$ if and only if it is a $U$-colimit diagram.

Proof. Apply Proposition 7.1.7.14 in the case $\operatorname{\mathcal{E}}= \Delta ^{0}$ (and use Example 7.1.7.3). $\square$