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\homework{Monads}

\begin{document}
\maketitle

\begin{exercise}
Prove that for any 2-category~$\cat{C}$ and any adjunction $\mo{f} \dashv \mo{g}$ in~$\cat{C}$, one can build a monad in~$\cat{C}$ whose underlying morphism is $\mo{f} \cocomp \mo{g}$.
\end{exercise}

\begin{exercise}
Prove that, in the 2-category~$\cat{CAT}$, for every monad~$\alg{M}$ with underlying functor $M$ on a category~$\cat{C}$ there is some adjunction $F \dashv U$ such that $M$ equals $F \cocomp U$.
Hint: use the underlying functor $U : \cat{Alg}(\alg{M}) \mto \cat{C}$ as the right adjoint.
\end{exercise}

\begin{remark}
The above theorem holds for monads in $\cat{CAT}$ but not necessarily for monads in other 2-categories.
\end{remark}

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