Kerodon

Remark 6.1.1.5 (Isomorphism Invariance). Let $\operatorname{\mathcal{C}}$ be a $2$-category, let $f,f': C \rightarrow D$ and $g,g': D \rightarrow C$ be $1$-morphisms in $\operatorname{\mathcal{C}}$, and let $(\eta , \epsilon )$ be an adjunction between $f$ and $g$. Suppose we are given invertible $2$-morphisms $\beta : g \xRightarrow {\sim } g'$ and $\gamma : f \xRightarrow {\sim } f'$. Let $\eta '$ denote the composition $\operatorname{id}_{C} \xRightarrow { \eta } g \circ f \xRightarrow [\sim ]{ \beta \circ \gamma } g' \circ f'$ and let $\epsilon '$ denote the composition $f' \circ g' \xRightarrow [\sim ]{ \gamma ^{-1} \circ \beta ^{-1} } f \circ g \xRightarrow {\epsilon } \operatorname{id}_{D}$. Then the pair $(\eta ', \epsilon ')$ is an adjunction between $f'$ and $g'$.