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I call $f=i\circ \overline{f}\circ \pi$ the canonical factorization of a function when I teach second year undergraduate discrete math (except that I write $f=\iota\circ \overline{f}\circ \nu$, using the Greek letters iota and nu for the inclusion map and the natural map). I have a handout for my students about this here (see the first link for the September 10 lecture).