Any function $\{(x_1,y_1), (x_2,y_2),... \}$ can be viewed as the union of "disjoint singleton" functions (I mean functions with distinct singleton co-domains, i.e. $\{x_1 \} \neq \{ x_2 \}$).
This reduction seems pleasing to me. Does it have a conventional uses or names?
Perhaps in the category of sets, where singleton sets are encoded as functions from the terminal element? Or elsewhere?
The categorical notions of presheaf and sheaf are an abstraction of the idea of constructing a function by gluing together smaller functions that are compatible in the sense that they agree whenever they are both defined. Your example arranges for the domains of the functions to be disjoint, so that they are automatically compatible. So an appropriate technical term would be "gluing": you are observing that any set-theoretic function can be obtained by gluing together singletons. See the Wikipedia page on the gluing axiom for some more information, in particular, how the presheaf and sheaf ideas pan out in situations where topological and algebraic considerations have to be taken into account.
