Subobject and quotient object as mutual-factorisation classes of monomorphisms and epimorphisms

Fix an object $C$ of a category $\mathcal{C}$. Two monomorphisms $m:A\to C$ and $n:B\to C$ (Monomorphism and epimorphism by left and right cancellation) mutually factor when there are morphisms $u:A\to B$ and $v:B\to A$ such that $m=n\circ u,n=m\circ v\mathrm{.}$ A subobject of $C$ is an equivalence class of monomorphisms into $C$ under mutual factorisation. The class represented by $m$ is denoted $[m]$. For representatives, write $[m]\le [n]$ when $m$ factors through $n$.

Dually, two epimorphisms $q:C\to Q$ and $r:C\to R$ mutually factor when $r=u\circ q$ and $q=v\circ r$ for suitable $u:Q\to R$ and $v:R\to Q$. A quotient object of $C$ is an equivalence class of epimorphisms out of $C$. Quotients are ordered by $[q]\le [r]$ when $r$ factors through $q$, the orientation dual to that for subobjects.

The equivalence-relation and representative-independence obligations are discharged by Mutual factorisation is an equivalence relation on monomorphisms into an object and dually on epimorphisms out of it ↗.

What $[m]$ is, and what it is not. The monomorphisms into $C$ generally form a proper class — already in $\mathbf{S}\mathbf{e}\mathbf{t}$, where every singleton admits a monomorphism into a one-point set — so $[m]$ is a class and not a set. Under this development's convention a class abbreviates a formula and is not an additional entity (Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why $\mathbf{C}\mathbf{A}\mathbf{T}$ is not formed), so $[m]$ is never a member of anything and the subobjects of $C$ are never gathered into a collection. Every statement written with the bracket notation below is shorthand for a statement about representatives: $[m]\le [n]$ means that $m$ factors through $n$, which the next two items show depends only on the two classes, and $[m]=[n]$ means that $m$ and $n$ mutually factor. Size conditions on subobjects are likewise stated on representatives later on this page, never by measuring a collection of classes.