The exercise is the following: Suppose $F$ has right adjoint $G$ and $H$ has right adjoint $J$ ($F:\mathbb{A}\rightarrow\mathbb{B}$ and $H:\mathbb{B}\rightarrow\mathbb{C}$ with $\mathbb{A,B,C}$ categories). Prove that $GJ$ is the right adjoint of $HF$.
What is the best way to start with? I want to do this by the two-way-rule, thus by bijective correspondence, but i don't see how to start. Someone who can help me?
Thank you very much.
$$\hom_{\Bbb A}(a,\ GJ(c))\, \simeq\, \hom_{\Bbb B}(F(a),\ J(c))\, \simeq\, \hom_{\Bbb C}(HF(a),\ c)\,.$$