homomorphisms are simply looser isomorphisms as far as their cohort of objects in their particular category are concerned.
Maybe “ideal” has to do with rings of these beasties and what these homomorphisms change into nothingness, or at least the zero of the ring. Oops, the math is beginning to get political.
"Loosen" things up and we go from isomorphisms to homomorphisms in a heartbeat. And it does not stop there.