Let $j: A \to B$ be a functor.
When $j$ has a left adjoint $L$, the codensity monad $\text{Ran}_jj$ will coincide with the monad $jL$. Since a left adjoint preserves all colimits, it is easy to observe the following:
Rem 1. Let $j$ be a functor with a left adjoint. Then if $j$ preserves a family of colimits, so does $\text{Ran}_jj$.
I hope that this is still true removing the assumption of having a left adjoint. Since my $j$ has values in a presheaf category, I wrote the end-formula, but I see no good reason for which the monad should preserve those colimits preserved by $j$.
Q1. Is the statement of Rem 1 still true removing the assumption of having a left adjoint? You can still assume that $j$ preserve all limits, if needed.
Q2. If it is false, are the some natural assumption that will make it true? (For example, $j$ fully faithful.)
