# Underlying functor U:K-Coalg → K-Mod preserves colimits.

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 '14 at 10:19

This comes simply from the observation that $U$ creates colimits and K-Mod has all colimits.
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 '14 at 9:38