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partial ordering on subobjects of an object
Let $\mathcal{C}$ be a category and $A$ an object $\mathcal{C}$. Recall that a subobject of $A$ is just an equivalent class of equivalent monomorphisms into $A$. A subobject is denoted by $[f:B\to A]$, where $f$ is monomorphic, or simply $[B]$, whenever there is no confusion. Furthermore, we write $B\subseteq A$ to mean that $B$ is a subobject of $A$, and write $B\cong C$ whenever $[B]=[C]$. The class of all subobjects of $A$ is denoted by ${\mathrm{Sub}}(A)$. We wish to put a partial order on ${\mathrm{Sub}}(A)$.
Given any two monomorphisms $f:B\to A$ and $g:C\to A$, define $g\leq f$ iff there is a morphism $h:C\to B$ such that
$\xymatrix{C\ar[dr]^{g}\ar[dd]_{h}\\ &A\\ B\ar[ur]_{f}}$ 
Now, if $f^{{\prime}}:B^{{\prime}}\to A$ is equivalent to $f$ and $g^{{\prime}}:C\to A$ is equivalent to $g$, then we have morphisms $x:C\to C^{{\prime}}$ and $x^{{\prime}}:C^{{\prime}}\to C$ and $y:B\to B^{{\prime}}$ and $y^{{\prime}}:B^{{\prime}}\to B$ such that the diagram below consisting of only the solid lines is commutative:
$\xymatrix{C\ar[dr]_{g}\ar[dd]_{h}\ar[rr]_{x}&&C^{{\prime}}\ar[dl]^{{g^{{\prime% }}}}\ar[ll]_{{x^{{\prime}}}}\ar@{.>}[dd]^{{h^{{\prime}}}}\\ &A&\\ B\ar[ur]^{f}\ar[rr]_{y}&&B^{{\prime}}\ar[ul]_{{f^{{\prime}}}}\ar[ll]_{{y^{{% \prime}}}}}$ 
We define $h^{{\prime}}=y\circ h\circ x^{{\prime}}$ (the dotted line above). Then the diagram above including the dotted line is commutative: as $f^{{\prime}}\circ h^{{\prime}}=f^{{\prime}}\circ(y\circ h\circ x^{{\prime}})=f% \circ(h\circ x^{{\prime}})=g\circ x^{{\prime}}=g^{{\prime}}$. Hence $g^{{\prime}}\leq f^{{\prime}}$. As a result, $\leq$ induces a binary relation on ${\mathrm{Sub}}(A)$:
$C\leq B$ 
iff there are representing monomorphisms $f:B\to A$ and $g:C\to A$ such that $g\leq f$. Let us use the same notation $\leq$ for the induced relation. Then $\leq$ on ${\mathrm{Sub}}(A)$ is a partial ordering on ${\mathrm{Sub}}(A)$:
 reflexivity:

$B\leq B$, because the composition of the identity morphism $1_{B}$ with any representing monomorphism $f:B\to A$ is $f$ itself.
 antisymmetry:

if $B\leq C$ and $C\leq B$, then there are representing monomorphisms $f:B\to A$ and $g:C\to A$ such that $f\leq g$ and $g\leq f$. So there are morphisms $h:B\to C$ and $h^{{\prime}}:C\to B$ such that
$\xymatrix{C\ar@<0.5ex>[dr]^{g}\ar@<0.5ex>[dd]^{{h^{{\prime}}}}\\ &A\\ B\ar@<0.5ex>[ur]_{f}\ar@<0.5ex>[uu]^{h}}$ is commutative. But this just means that $h$ and $h^{{\prime}}$ are inverses of one other, or $B\cong C$, so that they are the same subobject of $A$.
 transitivity:

if $D\leq C$ and $C\leq B$, then there are representing monomorphisms $f:B\to A$, $g:C\to A$ and $h:D\to A$ such that $h\leq g$ and $g\leq f$. This means there are morphisms $r:D\to C$ and $s:C\to B$ such that $g\circ r=h$ and $f\circ s=g$. Since $f\circ(s\circ r)=g\circ r=h$, $h\leq f$, and as a result, $D\leq B$.
Thus, $\leq$ turns ${\mathrm{Sub}}(A)$ into a partially ordered class, with top element $A$. With this, we may form the notions of unions and intersections of subobjects. Formally, let $\{B_{i}\mid i\in I\}$ be a collection of subobjects of $A$, indexed by a set $I$.

The union $C$ of $B_{i}$ is the supremum of the $B_{i}$’s with respect to $\leq$, provided that it exists. In notation, we write
$C=\bigvee_{{i\in I}}B_{i}.$ 
The intersection $D$ of $B_{i}$ is the infimum of the $B_{i}$’s with respect to $\leq$, provided that it exists. In notation, we write
$D=\bigwedge_{{i\in I}}B_{i}.$
For example, let $f:B\to A$ and $g:C\to A$ be subobjects of $A$. Then the pullback of $f$ and $g$, if it exists, is the intersection of $B$ and $C$.
Remark. It can be shown that in any Abelian category, ${\mathrm{Sub}}(A)$ under $\leq$ is a lattice for any object $A$.
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