As far as I can tell, the formulas you are writing arise from the usual duality between groups of multiplicative type and (fpqc sheaves of) finitely generated abelian groups.  You can find this in SGA3 volume 1.  Statement (**) seems to be some form of the standard fact that homomorphisms from Z to G form a group isomorphic to G.  This is a defining property of Z as a group.  You could probably make some kind of generalization to monoids using the natural numbers, but it's not clear how useful it would be.

It looks like Greg Stevenson basically answered your question in a comment, but you replied with something about groups in general, even though such groups didn't make an appearance in your question until the parenthetical comment at the end.  Could you make some effort to write your question in a more precise way, so we know what you want, and why you want it?  The fact that it came from your notes doesn't seem to be a satisfactory explanation for the fact that it is almost impossible to understand.