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.
Edit:
Let me add a comment to address the questions asked by Pace.
Is there a reason that $\nu$ is natural, but the factorization is canonical?
Natural and canonical mean different things. Natural means: determined by Nature. Canonical means: determined by the Canon (the law). Something becomes Canonical because it has been ruled to be so. The authority to call a concept Canonical might be the person who introduced the concept, or it might be the community who have used and developed the concept, but a canonical concept does not have to defend its naturality.
In mathematics, I try to restrict the use of the word Natural to situations where there is a natural transformation around, but I refer to the universal map of a set $S$ to a quotient set $S/E$ which maps $s\in S$ to its $E$-equivalence class $s/E$ as the ``natural map'' because much of the community uses that term (e.g., in the case where you map a group $G$ to a quotient group $G/N$ by mapping an element $g\in G$ to its coset $gN$).
Finally, to answer the question, I chose Natural for the quotient map because it is a common convention to use this word in this context. I prefer Canonical over Natural for the factorization $f=\iota\circ \overline{f}\circ \nu$ because, in this classroom setting, I prefer to avoid any confusion that might arise from two differing and new uses of the word Natural. (At least, I prefer some word that is different from Natural, and Canonical is grammatically correct.)
do your students find the term coimage palatable?
I never consider questions like this. A term is needed, and a correct/conventional term exists.