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Can anyone explain to me Yoneda Lemma proof in great details? i.e. they usually say " ... it is easy to see that these morphisms are inverse to each other.." without explanation.

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Perhaps it would be better if you explained what parts of the Yoneda lemma's proof you understand, and we can help with the bits that are unclear? If the only problem is understanding why the Yoneda embedding is fully faithful, there are two steps. For faithful, if a morphism $f$ is sent to a natural transformation $\eta$ in the functor category, we can recover $f$ by applying $\eta$ to the identity map. Therefore, no two different maps can be sent to the same transformation. For full, the definition of natural transformation implies that the image of the identity map determines everything. –  Aaron Dec 25 '11 at 9:24
thanks for the comment. –  user17090 Dec 25 '11 at 10:13
You would learn more if you try it yourself. –  Martin Brandenburg Jan 5 '12 at 13:37
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up vote 4 down vote accepted

See this text of our friend Zev Chonoles:

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Thank you very much. –  user17090 Dec 25 '11 at 10:09
Dear @AliBleybel: You're welcome! –  Pierre-Yves Gaillard Dec 25 '11 at 10:15
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