Exercise 6.1.1.6 (Functoriality). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $2$-categories. Suppose we are given $1$-morphisms $f: C \rightarrow D$ and $g: D \rightarrow C$ in $\operatorname{\mathcal{C}}$. Let $(\eta , \epsilon )$ be an adjunction between $f$ and $g$ in the $2$-category $\operatorname{\mathcal{C}}$, let $\eta '$ denote the composition
\[ \operatorname{id}_{ F(C) } \xRightarrow [\sim ]{} F( \operatorname{id}_{C} ) \xRightarrow { F(\eta ) } F( g \circ f ) \xRightarrow [\sim ]{\mu ^{-1}_{g,f} } F(g) \circ F(f), \]
and let $\epsilon '$ denote the composition
\[ F(f) \circ F(g) \xRightarrow [\sim ]{ \mu _{f,g} } F(f \circ g) \xRightarrow {F(\epsilon ) } F( \operatorname{id}_{D} ) \xRightarrow [\sim ]{} \operatorname{id}_{F(D)}, \]
where $\mu _{f,g}$ and $\mu _{g,f}$ are the composition constraints of the functor $F$ and the unlabeled isomorphisms are the identity constraints of $F$. Show that the pair $(\eta ', \epsilon ')$ is an adjunction between $F(f)$ and $F(g)$ in the $2$-category $\operatorname{\mathcal{D}}$.