Kerodon

Remark 6.2.1.8 (Composition of Adjoints). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $F': \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories which admit right adjoints. Then the composite functor $(F' \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ also admits a right adjoint. More precisely, if $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and $G': \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ are right adjoints of $F$ and $F'$, respectively, then the composite functor $(G \circ G'): \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is right adjoint to $(F' \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ (see Corollary 6.1.5.5).