Adjoints of two naturally isomorphic functors

Question

Suppose that $(F, G)$ and $(F', G')$ are two pairs of adjoint functors, and moreover that $F$ and $F'$ are two naturally isomorphic functors. It is true that $G$ and $G'$ are also naturally isomorphic?

Any thoughts would be appreciated.

Answer 1

Yes, and in fact each natural isomorphism $\alpha:F\to F'$ induces a canonical natural isomorphism $\beta:G'\to G$ defined as the composite $$G'\to GFG'\to GF'G'\to G,$$ where the maps are $\eta*G', G*\alpha*G',$ and $G*\varepsilon'$, $\eta$ being a choice of unit for the adjunction $F\vdash G$ and $\varepsilon'$ a choice of counit for the adjunction $F'\vdash G'$. We call $\beta$ the mate of $\alpha$.

A dual construction gives the mate of $\beta$, another natural transformation $F\to F'$. One checks using the triangle identities that the mate of $\beta$ is in fact $\alpha$ again; that is, the mate construction gives a bijection between the sets of natural transformations $F\to F'$ and of natural transformations $G'\to G$. Essentially because the mate of an identity is an identity, the mate constructions preserves natural isomorphisms, which gives the result you were interested in.