Asking for methods to produce the sum of natural numbers from the disjoint union of sets, it seems that the obvious way is to use the general definition, as coproduct, of the sum of sets. The accepted answer in that question suggested to use pairing with a "label", but the coproduct just needs a pair of "disjoint" injections, for example here in "Sets for Mathematicians", chapter 2:
Amusingly I have found a textbook in Spain, around 1974, that show the required injections.
(I found another one https://i.stack.imgur.com/j0QmF.jpg EDIT but in this case the author confirms me that the intention was just to show the idea of "increasing by one").
As you can see, it is very like the examples from Lawvere category of sets.
And now I am puzzled if the Spanish textbooks could have originated from some foreign, probably USA, material for primary school. Is this kind of representation, or drawing, used in New Math textbooks? Do you know any?