Categorical decorations of the emergent algebra sector, I

The emergent algebra sector of graphic lambda calculus is described in   arXiv:1305.5786  section 5 .  By a categorical decoration of the graphs from this sector I mean  representing such graphs as if the edges are arrows in a category. In this first post I sketch what I need, later I shall streamline everything, so don’t take anything at face value, but more with an exploratory, gaming attitude. Comments are welcomed.

The CO-COMM and CO-ASSOC moves are part of the moves which define this sector, along with the emergent algebra moves.  The first group of moves suggests to use a category with finite coproducts and with nontrivial cocommutative cogroupsThis would make things symmetric, because $\Gamma$ is a commutative group.

So we have a cocommutative group object $M$ with operation $\Upsilon: M \rightarrow M \vee M$, a commutative group object $\Gamma$  with operation $\cdot : \Gamma \times \Gamma \rightarrow \Gamma$ … and an operation $\delta: M \times G \rightarrow Hom(M,M)$, which suggests  we need a category with finite products and coproducts, with interesting commutative group and cocommutative cogroup objects.  The rules of decorations are the following (the first line tells that to the gate $\bar{\varepsilon}$ corresponds a composite of the $\delta$ operation with the evaluation $eval: Hom(M,M) \times M \rightarrow M$ . (Alternatively, the edge marked with “1” in the first diagram, along with M, may give an object (1,M), to explore.)

These arrows have to be compatible in the sense that they can be used to represent the emergent algebra moves. (Other constraints will follow.) For example, in the case of the R2 move the composition from the upper side of the figure have to be equal with the composition from the lower side of the figure:

I am interested in funny, crazy examples, (homotopy theory? combinatorics?), do you have any? If not yet, then wait for the sequel, or help me to clarify this subject.