For this to make sense in a given category$C$, we not only need a good notion of image. Note that it is not enough to have the image of $f\colon X \to Y$ as a subobject$\im f$ of $Y$; we also need to be able to interpret $f$ as a morphism from $X$ to $\im f$, 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 :