Let $(T, \eta, \mu)$ be monad over $\mathsf{C}$.
And let $\iota : \mathsf{Fix}(T) \hookrightarrow \mathsf{C}$ be the inclusion of the full subcategory of fixed points of $T$. By the universal property of Kan extensions we have a natural map $T \Rightarrow \mathsf{ran}_\iota \iota$ that is also a morphism of monads.
Q1: I wonder, when is such a map $T \Rightarrow \mathsf{ran}_\iota \iota$ an isomorphism?
The question is motivated by the following example. The fixed points of the ultrafilter monad are finite sets and yet the ultrafilter monad is precisely the codensity monad of those.
For the moment I have some partial answers. Call $j: \mathsf{Fix}(T) \hookrightarrow \mathsf{Alg}(T)$ the inclusion of the fixed points in the Eilenberg Moore category of algebras and call $F \dashv U$ the free forgetful adjunction. Observe that $\iota=Uj$.
Prop. Any of the following implies that $T \cong \mathsf{ran}_\iota \iota$.
$j$ is codense.
$\mathsf{ran}_j \iota = U$.
$\mathsf{ran}_\iota j = F$.
Q2: In some concrete example it might be easy to prove that $j$ is limit codense (this means that any object is a limit of some diagram in the image of $j$), do you know any criterion to promote a limit-codense subcatgory into a codense one?
Rem. Just a remark that might be relevant for some readers: any monad is the codensity monad of its forgetful functor.