Incidentally, this is part of a more general story about algebraic theories and relates to Tall-Wraith monoids (surprise, surprise).

Take an algebraic theory, say $V$, (which we identify with its category of models in Set) and a category $D$ with "sufficient structure". Then we can consider co-$V$-objects in $D$. These represent covariant functors $D \to V$. Let $H$ be such. Now if we take a co-$V$-object in $V$, say $B$, then by composition we get a covariant functor $D \to V \to V$. Under the "sufficient structure" assumption on $D$, representability is equivalent to having a left adjoint. As both $D \to V$ and $V \to V$ are representable, both have left adjoints. Thus their composition has a left adjoint and so is representable. Hence there is a co-$V$-algebra object representing $B_* H_*$ which we may as well write as $B \otimes H$. Lots of obvious naturality then implies that there is a corresponding bifunctor $VV^c \times DV^c \to DV^c$. In the particular case that $D = V$ we see that $VV^c$ is monoidal - which is the starting point of the construction of Tall-Wraith $V$-monoids - and more naturality then implies that the bifunctor $VV^c \times DV^c \to DV^c$ is an action of $VV^c$ on $DV^c$.

This generalises even further to give a - slightly odd-looking - action of $VV^c$ on the category of $V$-objects in $D$.

In the specific case in question, $V$ is the category of abelian groups and as $D$ is an abelian category, every object in $D$ is automatically a co-$V$-object in $D$.

(Bits of this story are in the Hunting of the Hopf Ring, other bits will be in a forthcoming paper with Sarah Whitehouse on Tall-Wraith monoids.)