Existence theorems for commutative diagrams

Given a relation $f\subset A\times B$, there exist two symmetric relations (see [1], Chapter 2) $f^{-1}f\subset A^2$, $ff^{-1}\subset B^2$. These relations make it possible to formalize definitions and proofs of existence theorems. For example, the equation $h=gf$, where $h$ and $g$ (or $h$ and $f$) are given maps, admits a solution $f$ ($g$, respectively) if and only if $hh^{-1}\subset gg^{-1}$ $(h^{-1}h\supset f^{-1}f)$. Well-known “homomorphism theorems” get more general interpretation. Namely, any map can be represented up to bijection as a composition of surjection and injection, and any morphism of diagrams can be represented up to isomorphism as a composition of epimorphism and monomorphism.
In this paper we further develop the scheme from [2] and consider it as an application in category of vector spaces and linear maps.