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The category '''Set''' with its subobject classifier satisfies the following property:
: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 {0, 1} = 2, denoted by 2<sup>''S''</sup>, 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 ''A'' → χ<sub>''A''</sub>, is a [[bijection]].
'''Axiom''': Given a category '''C''' with a subobject classifier, there exists an [[isomorphism]],
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