Kerodon

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Remark 7.1.7.6. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [d]^{U} \ar [r]^-{F} & \operatorname{\mathcal{C}}' \ar [d]^{U'} \\ \operatorname{\mathcal{D}}\ar [r] & \operatorname{\mathcal{D}}', } \]

where the horizontal maps are equivalences of $\infty $-categories. Then a morphism of simplicial sets $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-limit diagram if and only if $F \circ \overline{f}$ is a $U'$-limit diagram. Similarly, a morphism of simplicial sets $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a $U$-colimit diagram if and only if $F \circ \overline{g}$ is a $U'$-colimit diagram. This follows by combining Remark 7.1.6.8 with Corollary 4.6.4.18.