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Monoidal category. $\lambda_I = \rho_I : I\otimes I\to I$? If this equality can not be proved, in what categories it is false?

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This condition follows from the other axioms. I suggest you take a look at a beautiful J. Kock's paper "Elementary remarks on units in monoidal categories", which describes monoidal categories from a bit different (and more coherent) perspective.

However, proving the condition "by hand" may be a good exercise for you.

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  • $\begingroup$ Thank you. To be honest, I failed to do this exercise without peeking, though tried with at least 1 page of diagrams. :) In Kock's paper there are words: “Shortly after, it was shown by Kelly [6] that one of these four axioms for units in fact implies the three others. His proof constitutes nowadays the first three lemmas in many treatments of monoidal categories…” Can you recommend some treatments of monoidal categories (not written by Kelly, please :) )? @Mockup: $\endgroup$
    – beroal
    Jul 25 '11 at 0:21
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    $\begingroup$ For a good introductory text you may try the following: P. Selinger, "A survey of graphical languages for monoidal categories.". $\endgroup$ Jul 25 '11 at 17:55

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