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][1].

  [1]: http://math.colorado.edu/~kearnes/Teaching/Courses/F21/functions.pdf