I'm stuck with the next exercise. I don't know how can I solve it.
Let $\mathcal{F}:\mathcal{C}\to\mathcal{D}$ and $\mathcal{E},\mathcal{G}:\mathcal{D}\to\mathcal{C}$ three adjoint functors $\mathcal{E}\dashv\mathcal{F}\dashv\mathcal{G}$. Let $(\alpha,\beta)$ the unit and counit of the adjunction $\mathcal{E}\dashv\mathcal{F}$ and $(\eta,\varepsilon)$ the unit and counit of the adjunction $\mathcal{F}\dashv \mathcal{G}$. Suposse that the next conditions (1) holds:
a) $\text{Id}_{\mathcal{EFE}}=\mathcal{E}_{\alpha}\circ\beta_{\varepsilon}$
b) $\text{Id}_{\mathcal{FEF}}=\alpha_{\mathcal{F}}\circ\mathcal{F}_{\beta}$
Prove that the next conditions holds
a) $\text{Id}_{\mathcal{FGF}}=\mathcal{F}_{\eta}\circ\varepsilon_{\mathcal{F}}$
b) $\text{Id}_{\mathcal{GFG}}=\eta_{\mathcal{G}}\circ \mathcal{G}_{\varepsilon}$
First, I proved that the condition a) is equivalent to the condition b), thus, I only need to prove one of a) or b). Next, I wrote the commutative diagrams related to my hypothesis:
By (1) we have the next commutative diagrams
And because $\mathcal{E}\dashv\mathcal{F}$ and $\mathcal{F}\dashv\mathcal{G}$ we have the next pair of commutative diagrams
Thus I need to prove that the next diagram is commutative:
But I don't know how. By combining the previous diagrams, I obtained the next commutative diagram:
But in fact that diagram does not help. I saw that if I prove the previous diagram, then in fact I will prove that $\mathcal{F}_{\eta}$ is a natural isomorphism, but, again, I'm stuck. Any hint? I really appreciate any help you provide me.