I'm currently self-studying "Categories and Sheaves"by Schapira and Kashiwara, and I've been stuck on problem 1.19 all day today, so I was hoping that someone could help me out.
Let $\mathcal{C}$, $\mathcal{C'}$ be categories and $L_v:\mathcal{C} \rightarrow \mathcal{C'}$, $R_v:\mathcal{C'} \rightarrow \mathcal{C}$ be functors such that $(L_v,R_v)$ is a pair of adjoint functors (v=1,2). Let $\epsilon_v: id_\mathcal{C} \rightarrow R_v L_v$ and $\eta_v: L_v \circ R_v \rightarrow id_{\mathcal{C'}}$ be the adjunction morphisms. Prove that the two maps $\lambda$,$\mu$ : $$Hom_{Fct(\mathcal{C},\mathcal{C'})}(L_1,L_2) \rightleftarrows^{\lambda}_{\mu} Hom_{Fct(\mathcal{C'},\mathcal{C})}(R_2,R_1)$$ given by : $$\lambda(\varphi):R_2 \rightarrow^{\epsilon_1 \circ R_2} R_1 \circ L_1 \circ R_2 \rightarrow^{R_1 \circ \varphi \circ R_2} R_1 \circ L_2 \circ R_2 \rightarrow^{R_1 \circ \eta_2} R_1$$ for $\varphi \in Hom_{Fct(\mathcal{C},\mathcal{C'})}(L_1,L_2)$ $$\mu(\psi) : L_1 \rightarrow^{L_1 \circ \epsilon_2} L_1 \circ R_2 \circ L_2 \rightarrow^{L_1 \circ \psi \circ L_2} L_1 \circ R_1 \circ L_2 \rightarrow^{\eta_1 \circ L_2} L_2$$ for $\psi \in Hom_{Fct(\mathcal{C}',\mathcal{C})}(R_2,R_1)$ are inverse to eachother.
So far, I've been trying to play around with the zig-zag identity, but it doesn't seem to lead me anywhere. Any help will be greatly appreciated!