My question is, basically, the same as this one:
but I don't understand the given hint. Although I have constructed such $$φ:A(Fx,a)→X(x,Ga) \quad \text{by} \quad φ(h)=G(h)∘ϱx$$ and $$ψ:X(x,Ga)→A(Fx,a) \quad \text{by} \quad ψ(f)=ηa∘F(f)$$
and perhaps even can show that their composition $φψ = 1$ and hence $ψφ$ is idempotent, I don't understand how this proves that $εF⋅Fη$ is idempotent.
I would be very grateful for an explanation! Thanks in advance.
Since the Yoneda embedding is fully faithful, to prove that $\epsilon_{Fx}\circ F(\eta_x)$ is an idempotent endomorphism of $Fx$ it suffices to check that the induced natural transformation on the functor $A(Fx,\_)$ is idempotent. But by construction this induced natural transformation is precisely $\psi\varphi$.