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To categorize this notion, recall that, in category theory, a subobject is actually represented by a pair consisting of an object ''A'' and a [[monomorphism|monic arrow]] ''A → S'' (interpreted as the inclusion into another object ''S''). Accordingly, '''true''' refers to the element 1, which is selected by the arrow: '''true''': {0} → {0, 1} that maps 0 to 1. The subset ''A'' of ''S'' can now be defined as the [[pullback (category theory)|pullback]] of '''true''' along the characteristic function χ<sub>''A''</sub>, shown on the following diagram:
[[Image:SubobjectClassifier-01.
Defined that way, χ is a morphism ''Sub''<sub>C</sub>(''S'') → Hom<sub>C</sub>(S, Ω). By definition, Ω is a '''subobject classifier''' if this morphism χ is an isomorphism.
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