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The morphism ''χ<sub>j</sub>'' is then called the '''classifying morphism''' for the subobject represented by ''j''.
The above example of subobject classifier in Sets is very usefull because it enables us to easliy prove the following axiom:</i>
'''Axiom'''</i>
Given a category <math>\mathscr{C}</math>\mathscr{C}, then there exists an Isomorphisms</i>
<math>y:Sub_C(X)\cong Hom_C(X,\Omega)\hspace{.1in}\forall X\in C</math>
</i>
In Set this axiom can be restated as follows </i>
'''Axiom'''
The collection of all subsets of S denoted by <math>\mathcal{P}(S)</math>, and
the collection of all maps from S to the set <math>\{0,1\}=2</math> denoted by <math>2^S</math> are isomorphic i.e. the function <math>y:\mathcal{P}(S)\rightarrow2^S</math>,
which in terms of single elements of <math>\mathcal{P}(S)</math> is <math>A\rightarrow\chi_A</math>, it is a bijection.</i>
</i>
The above axiom implyes that a subobject calssifier can be defined as follows
'''Definition'''
<math>\Omega</math> is a
'''Subobject classifier''' iff there is a ``one to one" correspondence between subobject of X and morphisms
from X to <math>\Omega</math>.
== Further examples ==
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