# Equivalence of categories of coalgebras

I'm studying monadicity and comonadicity and I´m stuck with the following: Let $L\dashv R:X\rightarrow Y$ be an adjunction with unit $\eta$ and counit $\varepsilon$. The induced monad on $Y$ is $M=(RL,\eta,R\varepsilon L)$. Let $Y^{M}$ be the category of $M$-algebras and $F\dashv U:Y^{M}\rightarrow Y$ the free-forgetful adjunction.

Now let $C$ be the comonad on $X$ induced by $L\dashv R$ and $C'$ the comonad on $Y^{M}$ induced by $F\dashv U$. Let $X_{C}$ and $(Y^{M})_{C'}$ be the respective categories of coalgebras. Is it true that if $R$ is monadic then $X_{C}$ and $(Y^{M})_{C'}$ are equivalent? What I know so far is that the comparison functor $K:X\rightarrow Y^{M}$ induces, quite naturally, a functor $\overline{K}:X_{C}\rightarrow(Y^{M})_{C'}$ but I can´t get past this. I'll appreciate any help with this, thanks.

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Have you looked at your favorite example of an adjunction? – Martin Brandenburg Mar 14 '13 at 4:25
If $R$ is monadic then you may as well assume $X = Y^M$, but that does somewhat trivialise the question. So perhaps what you should do is show that $C$ and $C'$ are equivalent in a suitable sense. – Zhen Lin Mar 14 '13 at 8:37
Hello user66685, welcome to Mathematics.SE . I would suggest you take a moment to think about a more personalized nickname than the automatic one you have been assigned – magma Mar 14 '13 at 12:09