I have some questions about generalizations of abelian groups, relative to symmetric monoidal categories.

1) Let $C$ be a cocomplete cartesian monoidal category with equalizers. I can show that the forgetful functor $U : \mathsf{Ab}(C) \to C$ has a left adjoint $F$. The idea is to write $U$ as the composition $\mathsf{Ab}(C) \to \mathsf{CMon}(C) \to C$, the second functor has a left adjoint (symmetric algebra), and the first functor has a left adjoint which is a generalization of the Grothendieck construction (should I explain it?). Is there are more direct description $F$? Also I would like to know if the corresponding monad $T : C \to C$ preserves reflexive coequalizers, because then $U$ is monadic, which would be useful.

2) Let $C$ be a cocomplete symmetric monoidal category (perhaps assumed to be presentable). I would like to extend 1) to the category $\mathrm{AbHopf}(C)$ of cocommutative commutative Hopf monoids in $C$, i.e. the category of abelian group objects in the cartesian monoidal category $\mathrm{CoMon}_c(C)$ of cocommutative comonoids in $C$. In particular, I would like to know if $\mathrm{AbHopf}(C) \to C$ is monadic, if the corresponding monad has a nice description and if it preserves reflexive coequalizers. Obviously it would be helpful to understand $\mathrm{CoMon}_c(C) \to C$ first.

There are some papers by Porst, Barr, Takeuchi and others about these sort of questions, but I haven't found an answer there. Actually my goal is to endow $\mathrm{AbHopf}(C)$ with a tensor product such that the left adjoint $C \to \mathrm{AbHopf}(C)$ is symmetric monoidal.