Re: first isomorphism theorem

Toby Bartels pointed out

...

the essence of the isomorphism theorem is that the forgetful functor to sets not only preserves limits but also preserves images and coimages even though it does not preserve coequalizers and cokernel pairs.

In fact, as recent papers of Adamek,Lawvere,& Rosicky, and also of Pedicchio & Wood, have exploited, such forgetful functors do preserve all coequalizers of pairs which admit a reflexivity in the category; the preservation of images is a consequence.