The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\mathbf E$). We can also define the category of algebras for an endofunctor on $\mathbf E$.

1. Is there a characterisation of those functors $u : \mathbf B \to \mathbf E$ which are the forgetful functors from the category of algebras for some endofunctor on $\mathbf E$ (up to an equivalence over $\mathbf E$)?

Such forgetful functors do not always admit left adjoints. However, when they do, they are in particular monadic (the conditions of the monadicity theorem being easy to verify). So we can expect such a characterisation to imply that $u \colon \mathbf B \to \mathbf E$ creates $u$-split coequalisers. Furthermore, when $u$ is monadic, the induced monad is algebraically-free (by definition). So (in the presence of a left adjoint) my question can really be seen as asking when a monadic functor induces an algebraically-free monad.

This motivates my second question.

2. Is there an intrinsic characterisation of those monads on $\mathbf E$ that are algebraically-free?

By "intrinsic", I mean in terms of the data $(T, \mu, \eta)$, without reference to the forgetful functor from the category of algebras.

These seem natural questions, and I expect there are answers in the literature somewhere, but so far I have not been successful in finding them.

The monadicity theorem characterises when a functor $u : \mathbf B \to \mathbf E$ is the forgetful functor from the category of algebras for some monad on $\mathbf E$ (up to an equivalence over $\mathbf E$). We can also define the category of algebras for an endofunctor on $\mathbf E$.

1. Is there a characterisation of those functors $u : \mathbf B \to \mathbf E$ that are the forgetful functors from the category of algebras for some endofunctor on $\mathbf E$ (up to an equivalence over $\mathbf E$)?

Such forgetful functors do not always admit left adjoints. However, when they do, they are in particular monadic (the conditions of the monadicity theorem being easy to verify). So we can expect such a characterisation to imply that $u \colon \mathbf B \to \mathbf E$ creates $u$-split coequalisers. Furthermore, when $u$ is monadic, the induced monad is algebraically-free (by definition). So (in the presence of a left adjoint) my question can really be seen as asking when a monadic functor induces an algebraically-free monad.

This motivates my second question.

2. Is there an intrinsic characterisation of those monads on $\mathbf E$ that are algebraically-free?

By "intrinsic", I mean in terms of the data $(T, \mu, \eta)$, without reference to the forgetful functor from the category of algebras.

These seem natural questions, and I expect there are answers in the literature somewhere, but so far I have not been successful in finding them.