Let $$j: A \to B$$ be a fully faithful functor.

When $$j$$ has a left adjoint $$L$$, the codensity monad $$\text{Ran}_jj$$ will coincide with the monad $$jL$$ and thus will be idempotent, because $$A$$ is reflective in $$B$$.

Rem 1. Let $$j$$ be a fully faithful functor with a left adjoint. Then the codensity monad $$\text{Ran}_jj$$ is idempotent.

I was wondering if this is still true removing the assumption of having a left adjoint.

In a relatively trivial way, one can reformulate what I said in the following way.

Rem 2. Let $$j$$ be a fully faithful functor. TFAE:

• $$\text{Ran}_jj$$ preserve itself.
• $$\text{Ran}_jj$$ is idempotent.

So, the question is finally the following,

Q1. Let $$j$$ be a fully faithful functor, is it true that one of the two equivalent conditions in Rem 2 is verified? You can still assume that $$j$$ preserve all limits.

Q2. If not, are there some natural assumptions that will make it true?

Let $$j: A \to B$$ be the inclusion $$Set_{\leq 3} \to Set$$ of sets of cardinality $$\leq 3$$ into $$Set$$. Then the codensity monad for $$j$$ is the ultrafilter monad (as I learned from Tom Leinster-- I think this observation goes back to Isbell). This monad is not idempotent.