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the category of right comodule of coalgebra is a monoidal category according the following

the associativity constraint is defined as a_{U,V,W}((u\otimes v)\otimes w)=u_0\otimes (v_0\otimes w_0)\Phi(u_1\otimes v_1\otimes w_1), where the coation is \delta(v)=\sum v_0\otimes v_1.

How can we check the Pentagon axiom according the above definition?

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Your question seems to be missing the definitions of some of your notation. – S. Carnahan May 14 '13 at 1:51
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Please look over mathoverflow.net/howtoask and revise this question: it is missing definitions, should have better capitalization and punctuation, and might as well have the mathematics typeset as actual TeX (see the box "How to write math" on the right-hand column). But probably no change will fix your question, as in general there is no good monoidal structure for the category of comodules of a general coalgebra. Two situations in which comodule categories are monoidal are when the coalgebra in question is cocommutative, or when the coalgebra is given a Hopf algebra structure. – Theo Johnson-Freyd May 14 '13 at 4:13

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