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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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    For a good introductory text you may try the following: P. Selinger, "A survey of graphical languages for monoidal categories.".2011-07-25