If $T$ is a monad on a category $\mathcal C$ and $T'$ is a monad on $T$-algebras, then (if I understand the answers of this question correctly) the adjunction between $\mathcal C$ and $T'$-algebras is not necessarily monadic.
My question is: what happens if we make some additional assumption on $T'$? More precisely: let $\mathcal O$ be a (set-theoretic, non-symmetric) operad (typically, the "free monoid" operad), and assume that $T$-algebras come equipped with a biclosed monoidal structure. Is it always true that the adjunction between $\mathcal C$ and $\mathcal O$-algebras in $T$-algebras is monadic?
