For this to make sense in a given category , we not only need a good notion of image. Note that it is not enough to have the image of as a subobject of ; we also need to be able to interpret as a morphism from to , because it is this morphism that we are asking to be an isomorphism.
One general abstract way to define an embedding morphism is to say that this is equivalently a regular monomorphism.
If the ambient category has finite limits and colimits, then this is equivalently an effective monomorphism. In terms of this we recover a formalization of the above idea, that an embedding is an iso onto its image :