subobjects up to automorphisms?

Stream: maths

[Scott Morrison]

A question purely about mathematical nomenclature.

There's an equivalence relation on subobjects i : A ↪ X of fixed object X in some category, given by i ~ j iff there is an automorphism φ of X so i ≫ φ = j.

Does this have a standard name at that level of generality?

(e.g. the equivalence classes of subobjects of the vector space ℝⁿ are given by the rank.)

[Kevin Buzzard]

A characteristic subgroup of a group is a subgroup fixed by all automorphisms. Is the concept of being a characteristic subgroup actually studied much? The name always struck me as odd. Does this equivalence relation come up in maths? In the example you describe the equivalence relation is equal to " we are abstractly isomorphic, forget the superspace" but of course in group theory this will not be the case. I don't know a name.