expressed as an equalizer of two morphisms) and a "quotient object" to be any equivalence class of "regular epimorphisms" (morphisms which can be expressed as a coequalizer of two morphisms)

Interpretation

This definition corresponds to the ordinary understanding of a subobject outside category theory. When the category's objects are sets (possibly with additional structure, such as a group structure) and the morphisms are set functions (preserving the additional structure), one thinks of a monomorphism in terms of its image.

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