I've started reading Fuhrmann's A Polynomial Approach to Linear Algebra and have just seen the canonical factorization of functions:
Given a function $f:A\to B$, the canonical factorization is $f=f_3\circ f_2\circ f_1$ where $$A\stackrel{f_1}\longrightarrow A/R \stackrel{f_2}\longrightarrow f(A) \stackrel{f_3}\longrightarrow B \\ f_1(a) = \{x\mid x\in A, f(x)=f(a)\} =: A_a \\ f_2(A_a) = f(a) \\ f_3(b) = b$$
I can see that $f_1$ is surjective, $f_2$ is bijective, and $f_3$ is injective. However, while it is interesting that a function can be decomposed this way, I don't see any immediate application of this. What is this factorization useful for?