(Co-)limit characterization of subobjects mapping to subobjects? My motivating example is the category of rings, (though this of course works the same for groups). Given a homomorphism $\phi : A \rightarrow B$, and a subring of A, given by the inclusion $ \iota : A' \hookrightarrow A $, the image of A' is always a subobject of B, that is, there exists an inclusion $j : \phi(A') \hookrightarrow B$.
For preimages of subrings of B in the context of the same homomorphism, its a little easier for me to comprehend, because we start with the diagram $A \xrightarrow{\phi} B \xleftarrow{\iota} B'$, so for the preimage of $B'$ being a subobject of A, we simply need the pullback of the diagram.
Is there a way of characterizing the property of subobjects of $A$ being mapped to subobjects of $B$ under their image using a (co-)limit construction? If so, what is the name of the construction?
 A: The image of a morphism is a subobject by definition, so the question could be given two senses: why is the set-theoretic image also a ring-theoretic image, or why do ring-theoretic images exist?
The existence of images for morphisms is essentially the definition of a regular category. Rings form a regular category because they're monadic over sets-if you're not familiar with monads, all algebraic structures and many besides are monadic. This answers the second question: rings have enough finite limits and sufficiently stable regular epimorphisms to guarantee images. The answer to the first is similar: forgetful functors out of monadic categories automatically preserve all limits and split coequalizers, and in particular (co)images*, the coequalizers of kernel pairs of morphisms
*The coimage coincides with the image, i.e. the equalizer of the cokernel pair, in any regular category-I think the classical cases of this are called the first isomorphism theorem. Anyway it's unclear whether one is really thinking of the image or the coimage when doing classical abstract algebra.
