Are commuting squares preserved by taking adjoints?

Question

Suppose $A\rightarrow A'$ in category C and $B\rightarrow B'$ in category D, and you have a commutative square, the top and bottom arrows of which are $FA\rightarrow B$ and $FA'\rightarrow B'$, repsectively. When you pass to the adjoint, does the resulting square $A\rightarrow GB$ and $A'\rightarrow GB'$ also commute?

Answer 1

Yes. If $f:A\to A'$, $g:B\to B',h:FA\to B,k:FA'\to B',$ then the transpose of $kF(f):FA\to B'$ is $\bar k f:A\to GB'$ and the transpose of $gh$ is $G(g)\bar h:A\to GB'$. Thus $kF(f)=gh$ if and only if $\bar k f=G(g) \bar h$. This is what it means to say the bijection $\hom(FX,Y)\cong\hom(X,GY)$ is natural in $X$ and $Y$.