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I am stuck with the following theorem:

Theorem : Let $K$ be a commutative ring, and let $U \colon K\mathrm{-Coalg} \to K\mathrm{-Mod}$ be the underlying functor from the category of $K$-coalgebras $K\mathrm{-Coalg}$ to the category of $K$-modules $K\mathrm{-Mod}$. Then $U$ preserves colimits.

Could you guys help me to prove this theorem (with out assuming that $U$ has a right adjoint)?

Dan

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btw, nobody would call this a "Theorem". –  Martin Brandenburg Jan 18 at 10:19

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This comes simply from the observation that $U$ creates colimits and K-Mod has all colimits.

(More generally, any comonadic functor creates colimits)

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Could you guys explain more how can U create colimits since I have no idea about comonadic functor. Thanks, Dan –  Dan Jan 19 at 7:33
    
You don't have to know comonadic functors. You only have to know the definitions. What have you tried? It's straight forward to check that $U$ creates colimits. Just dualize the usual proof (see Mac Lane etc.) that the forgetful functor from algebras to modules creates limits. –  Martin Brandenburg Jan 19 at 9:38

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