Proving that the transformation obtained from an adjoint pair is natural

I am reading Homological Algebra by J.J. Rotman and am unable to do this problem. Given an adjoint pair $(F,G)$ where $F : \mathcal{C} \to \mathcal{D} $ and $G : \mathcal{D} \to \mathcal{C} $ are two covariant functors, we can obtain a natural transformation $ \eta : \mathbb{1_{\mathcal{C}}} \to GF $. I understand the definition of $\eta$ but am unable to prove that the transformation is natural. Couls someone please help me prove that for $f \in Hom (C,C')$, $GF(f)\eta_C=\eta_{C'}f$ ? Thanks !

Solution 1:

By Yoneda, $\hom(Fx,-) \to \hom(x,G(-))$ is completely determined by a morphism $x \to G(F(x))$. But since $\hom(Fx,-) \to \hom(x,G(-))$ is natural in $x$, also $x \to G(F(x))$ is natural in $x$.