Kerodon

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Proposition 7.1.7.14 (Transitivity). Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $V: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories.

$(1)$

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

Proof. Apply Remark 7.1.6.5. $\square$