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I am looking for a nontrivial example of the following:

Let a monoid $A$ be given with unit $e$, and two of its distinguished disjoint submonoids $B_1$ and $B_2$ (s.t. $B_1\cap B_2=\{e\}$), endowed with monoid homomorphisms $\Delta_i:A\to B_i$ such that $\Delta_i|_{B_i}={\rm id}_{B_i}\ $ ($i=1,2$) . Moreover, also let a mapping $\sigma:A\to A$ be given, such that for all $a\in A$: $$\sigma(\Delta_1(a))\cdot\Delta_2(a)=e= \Delta_1(a)\cdot\sigma(\Delta_2(a)) $$ In particular, each element of $B_1$ is right invertible, and each element of $B_2$ is left invertible in $A$.

Such an example is trivial if $B_1=B_2=A$.

Motivation: I am looking for Hopf-like monoids in cartesian categories, in which the counit axioms for the unique $A\to 1$ morphism are not posed. Then, the comultiplication map $\Delta:A\to A\times A$ has to be of the form $\Delta=(\Delta_1,\Delta_2)$ and coassociativity means ${\Delta_i}^2=\Delta_i$ and $\Delta_1\Delta_2=\Delta_2\Delta_1$. $B_1\cap B_2=\{e\}$ ensures this latter equality. Of course, $\sigma$ would be the antipode.

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