first isomorphism theorem

Is there a suitable first isomorphism theorem in category theoretic language? One barrier for me seems to be the correct notion of an image. What seems best right now would be that given f:X->Y, ker(coker f) = coker ( ker f). Here the left hand side would represent im f, and the right hand side X/kerf. If this is the right notion, what are the necessary conditions on the category for it to hold? -Andrei Prokopiw