Let $\mathcal{C}$ and $\mathcal{D}$ be two categories and $F$ and $G: \mathcal{C}\to \mathcal{D}$ be two functors. Suppose $F$ and $G$ have right adjoints $F^{\wedge}$ and $G^{\wedge}: \mathcal{D}\to \mathcal{C}$.


Now let $T:F\Rightarrow G$ be a natural transformation.



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Does a natural transformation of functors induce a natural transformation between their right adjoints?
I should have made my comment an answer earlier. The answer is yes:


Let $\eta: 1 \to F^\wedge F$ denote the unit of the adjunction and $\varepsilon: G G^\wedge \to 1$ the counit. Then form the composite


$$G^\wedge \stackrel{\eta G^\wedge}{\to} F^\wedge F G^\wedge \stackrel{F^\wedge TG^\wedge}{\to} F^\wedge G G^\wedge \stackrel{F^\wedge \varepsilon}{\to} F^\wedge.$$


