Many years ago, Lawvere showed that the forgetful functor $U: \mathbf{Endo}\to \mathbf{Set}$ has a left adjoint $F$ if and only if $\mathbf{Set}$ has a natural numbers object, where $\mathbf{Endo}$ is
the category of endos $f: A\to A$, where $A$ is a set. Then $T=UF$ is the
functor part of a monad $(T,\mu,\eta)$. I learned some years ago too that $\mathbf{Endo}$ satisfies
the Beck conditions in Beck's Theorem, so that the category of $\mathbf{Set}^T$ of
$T$-algebras is equivalent to $\mathbf{Endo}$. 

An exercise in Barr and Wells states that under certain conditions, the equivalence
guaranteed by Beck's Theorem is in fact an isomorphism (see Exercise PPTT p. 116).
My question is: whether, in the case of the monad $(T,\mu,\eta)$ mentioned above,
$\mathbf{Endo}$ turns out to be *isomorphic* to $\mathbf{Set}^T$? --- either because the conditions
mentioned in the exercise are satisfied or for some other reason.