Corollary 6.1.5.5. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing objects $C$, $D$, and $E$, together with $1$-morphisms
\[ f: C \rightarrow D \quad \quad g: D \rightarrow C \quad \quad f': D \rightarrow E \quad \quad g': E \rightarrow D. \]
If $f$ is left adjoint to $g$ and $f'$ is left adjoint to $g'$, then $f' \circ f$ is left adjoint to $g \circ g'$.