What should I write on Weibel exercise A.6.3.?

Question

For left $(L,R):A\to B$ adjoint, with $\epsilon : id_{A} => RL$ and $\eta:LR => id_{B}$, Show $\eta(RL) \circ \eta = (RL \eta) \circ \eta$ and $\epsilon \circ (LR \epsilon)= \epsilon \circ(\epsilon LR)$.

For this problem what should I write on? I think the diagram chasing may be needed, but it is still clear that they're natural isomorphism. What should I write down for formal proof?

Answer 1

Just use naturality: $\eta:\text{id}\to RL$ so $\eta_y\circ f=RL f\circ\eta_x$ for any $f:x\to y$. In particular take $f=\eta_x,y=RLx$, to get $\eta_{RLx}\circ \eta_x=RL\eta_x\circ\eta_x$.The same story goes for $\epsilon$.