# Equivalent definitions of adjunction morphisms

I am struggling with the following exercise from Emily Riehl's Category Theory in Context, regarding adjunction morphisms: paraphrasing, let $$F : C \to D, G: D \to C$$ and $$F' : C' \to D', G' : D' \to C'$$ be functors such that $$F \dashv G$$ and $$F' \dashv G'$$. We now consider functors $$H : C \to C'$$ and $$K : D \to D'$$ such that the squares of adjunctions commute, that is so that $$HG = G'K$$ and $$KF = F'H$$. The exercise then asks to prove that the following are equivalent:

1. if $$\eta, \eta'$$ are the units of the adjunctions, then $$H \eta = \eta' H$$, i.e. $$H\eta_c = \eta'_{Hc}$$.
2. if $$\epsilon, \epsilon'$$ are the counits of the adjunctions, then $$K\epsilon = \epsilon'K$$.
3. the arrow compositions $$D(Fc,d) \xrightarrow{\simeq} C(c,Gd) \xrightarrow{H} C(Hc,HGd) = C(Hc,G'Kd)$$ and $$D(Fc,d) \xrightarrow{K} D(KFc,Kd) = D(F'Hc,Kd) \xrightarrow{\simeq} C(Hc, G'Kd)$$ are equal.

I've though about this for a fair amount of time, trying to use the adjunction relations between transposes/adjunts and the commutativity relations, with no luck: the only implication I have managed to prove is $$(3) \Rightarrow (1)$$, via tracking $$1_{Fc}$$ along both arrows, which give $$H\eta_c$$ and $$\eta'_{Hc}$$ respectively.

Any hints?

• Write out the isomorphisms of homsets explicitly in terms of units and counits. – Derek Elkins Jan 29 at 13:23
• @DerekElkins Thanks for the idea, I think I was using (co) units as black boxes a bit too much. I believe I have managed to conclude the result via your hint, and would really appreciate a sanity check. Cheers. – Guido A. Jan 29 at 18:44

An adjunction $$F\dashv G$$ with $$F:C\to D$$ can be represented as a category $$\mathcal F$$ containing (disjoint isomorphic copies of) $$C$$ and $$D$$ with a further arrow $$\tilde\delta:c\to d$$ for each $$\delta:Fc\to d$$.
Due to the adjunction, $$C$$ is a coreflective full subcategory, while $$D$$ is a reflective full subcategory of $$\mathcal F$$.
Now the conditions imply that $$H$$ and $$K$$ induces a functor $$\mathcal H:\mathcal F\to\mathcal F'$$, and both maps in 3. describe the action of this functor on the newly added homsets.
Following Derek Elkins' hint, I think I have managed to solve the problem: first off, by noting that the commutative diagram of $$(3)$$ is equivalent to the commutative diagram with the isomorphism legs reversed, from the technique used to solve $$(1) \iff (3)$$ one can similarly prove $$(2) \iff (3)$$.
Having said that, $$(1) \iff (3)$$ follows from the fact that the transpose of an arrow $$f : Fc \to d$$ can be defined (or characterized) as the composition $$\eta_cGf$$. Suppose that $$(1)$$ holds and let $$f : Fc \to d$$ be an arrow. Now, commutativity of the diagram amounts to showing $$H(\eta_cGf) = \eta'_{Hc}G'Kf \tag{\star}$$ since the transpose of $$Kf : F'Hc = KFc \to d$$ is, via the same remark, $$\eta'_{Hc}G'Kf$$. In effect, $$H(\eta_cGf) = H\eta_c HGf \stackrel{(1)}{=} \eta'_{Hc}HGf = \eta'_{Hc}G'Kf$$ as desired.