My first question here.

Accordingly to M. Barr "Coequalizers and free triples" by a free triple (or free monad) generated by an endofunctor $R: X\rightarrow{X}$ we mean a triple $T=(T,\eta,\nu)$ and a natural transformation $p: R\rightarrow{T}$ such that if $T= (T',\eta',\nu')$ is another triple and $p': R\rightarrow{T'}$ is a natural transformation, then $p'=\tau{p}$, where $\tau:T\rightarrow{T'}$ is a map of triples.

On the other hand we have always a monad in the form $(GF,\eta,G\epsilon{F})$ when we have an adjunction $F\dashv{G}$ where $\eta$ is the unit and $\epsilon$ de counit of the adjunction.

Can the endofunctor $R$ have the form of an adjunction (for exemple between a free and a forgetful functor) in the first definition? When does it happen and which is the relationship between the adjunction and the free monad after all?