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We found one dictionary that defines the word classifying morphism:

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  1. classifying morphism: Wiktionary

Definitions from Wiktionary (classifying morphism)

noun:  (category theory) A morphism from an object to the subobject classifier which corresponds to a unique subobject of the said object, which subobject is the pullback, along this morphism, of the "true" global element of the subobject classifier.

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