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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 !

Najib Idrissi
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user90041
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1 Answers1

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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 $x$.

Martin Brandenburg
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    Thanks for your answer, however I did not understand the second line. Are we using some lemma here ? – user90041 Jun 16 '14 at 10:16
  • I brushed up my knowledge of Yoneda lemma and related stuff, I think I understand it now. Thanks really a lot ! – user90041 Jun 16 '14 at 10:22