Equivalent definitions of adjunction morphisms

adjoint-functorscategory-theory

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?

Best Answer

Hint with another perspective:

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.

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