nLab
monadic functor

Given a pair of adjoint functors $F:C\to D:U$, $F⊣U$, with unit $\eta :{\mathrm{Id}}_C\to U\circ F$ and counit $ϵ:F\circ U\to {\mathrm{Id}}_D$, one constructs a monad $T=(T,\mu ,\eta )$ setting $T=U\circ F:C\to C$, $\mu =UϵF:TT=UFUF\to UF=T$. Consider the Eilenberg–Moore category $C^T$ of $T$-algebras ($T$-modules) in $C$. Clearly $U({ϵ}_M):TUM=UFUM\to UM$ is a $T$-action. In fact there is a canonical comparison functor $K^T:D\to C^T$ given on objects by $K(M)=(UM,U({ϵ}_M))$. We then say that we have a monadic adjunction.

A functor $U:D\to C$ is monadic if it has a left adjoint $F:C\to D$ and the comparison functor $K^T:D\to C^T$ is an equivalence of categories. In other words, up to equivalence, monadic functors are precisely the forgetful functors defined on Eilenberg–Moore categories for monads. A category $D$ is monadic over a category $C$ if there is a functor $U:D\to C$ which is monadic.

Various versions of Beck’s monadicity theorem (old-fashioned name of some schools: tripleability theorem) give sufficient, and sometimes necessary, conditions for a given functor to be monadic.