Let $\operatorname{Mono}(-,B)$ be the class of all monomorphisms into object $B$ in $\mathcal{C}$. Two elements $f\colon A_{1}\to B$ and $g\colon A_{2}\to B$ in $\operatorname{Mono}(-,B)$ are equivalent if there exist two morphisms $r:A_{1}\rightarrow A_{2}$ and $s:A_{2}\rightarrow A_{1}$ such that we have the following two commutative diagrams:

$\xymatrix@R-=2pt{A_{1}\ar[dr]^{f}\ar[dd]_{r}\\ &B\\ A_{2}\ar[ur]_{g}}\xymatrix@R-=2pt{&&\\ &and&\\ &&}\xymatrix@R-=2pt{A_{1}\ar[dr]^{f}\\ &B\\ A_{2}\ar[ur]_{g}\ar[uu]^{s}}$

It is easy to see that both $r$ and $s$ are monomorphic. In fact, $A_{1}$ and $A_{2}$ are isomorphic objects, as $f=g\circ r$ and $g=f\circ s$, so that $f=(f\circ s)\circ r$, or $1_{{A_{1}}}=s\circ r$, since $s\circ r$ is monomorphic. Similarly $1_{{A_{2}}}=r\circ s$. Monomorphism equivalence is an equivalence relation on $\operatorname{Mono}(-,B)$.

Definition. A subobject of an object $X$ in $\mathcal{C}$ is an equivalence class in $\operatorname{Mono}(-,X)$. Let’s write

for a subobject of $X$, and a monomorphism $f:A\to X$ a representative of $[A\to X]$. If there is no danger of confusion, it is often easier to identify a subobject $[A\to X]$ by $A$, and simply write

as long as we keep in mind that, along with the object $A$, there is a monomorphism from $A$ to $X$. The class of subobjects of $X$ shall be denoted by

Definition. A category $\mathcal{C}$ is said to be well-powered or locally small if for every object $X$ in $\mathcal{C}$, the class ${\mathrm{Sub}}(X)$ is a set. Most common categories are locally small.

Suppose now that $\mathcal{C}$ is well-powered and has pullbacks (a pullback exists for every pair of morphisms into the same object). We shall turn ${\mathrm{Sub}}$ into a functor from $\mathcal{C}$ to Set. For every morphism $\alpha:X\to Y$, define ${\mathrm{Sub}}(\alpha)$ as follows:

take a representative $g\in[B\to Y]$, consider the pullback of $\alpha$ and $g$ indicated in the commutative diagram below:

$\xymatrix@C+=30pt@R+=40pt{A\ar[d]_{f}\ar[r]^{{\beta}}&B\ar[d]^{g}\\ X\ar[r]^{{\alpha}}&Y.}$

Since $g$ is monomorphism, so is $f:A\to X$, and hence $A$ is a subobject of $X$. We set ${\mathrm{Sub}}(\alpha)([g]):=[f]$.

Because the diagram is a pullback, ${\mathrm{Sub}}(\alpha)([g])$ gives a unique value, and thus

is a well-defined morphism. Furthermore, it is easily verified that ${\mathrm{Sub}}(1_{X})=1_{{{\mathrm{Sub}}(X)}}$ and ${\mathrm{Sub}}(\alpha\circ\beta)={\mathrm{Sub}}(\beta)\circ{\mathrm{Sub}}(\alpha)$. Thus, ${\mathrm{Sub}}$ is a contravariant functor from $\mathcal{C}$ to Set (or a covariant functor $\mathcal{C}^{{op}}\to\textbf{Set}$), and is called the subobject functor of $\mathcal{C}$.

Dually, given an object $A$ in a category $\mathcal{C}$, we can define an equivalence relation on $\operatorname{Epi}(A,-)$, the class of all epimorphisms from $A$, by reversing all arrows in the previous paragraph. Specifically, two elements $f\colon A\to B_{1}$ and $g\colon A\to B_{2}$ in $\operatorname{Epi}(A,-)$ are equivalent if there exist two morphisms $B_{1}\rightarrow B_{2}$ and $B_{2}\rightarrow B_{1}$ such that the following two diagrams commute:

$\xymatrix@R-=2pt{&B_{1}\ar[dd]\\ A\ar[ur]\ar[dr]\\ &B_{2}}\xymatrix@R-=2pt{&&\\ &and&\\ &&}\xymatrix@R-=2pt{&B_{1}\\ A\ar[ur]\ar[dr]\\ &B_{2}\ar[uu]}$

Definition. A quotient object of $X$ is an equivalence class in $\operatorname{Epi}(X,-)$. A typical quotient object is denoted by $[X\to B]$. If $\mathcal{C}$ is a small category and has pushouts, then there is a covariant functor ${\mathrm{Quo}}:\mathcal{C}\to\textbf{Set}$ taking each object of $\mathcal{C}$ to its set of quotient objects and each morphism between two objects to a morphism between the sets of their quotient objects. ${\mathrm{Quo}}$ is called the quotient object functor of $\mathcal{C}$.