By the way, I see the following from [Ellerman (2016) "Brain Functors: A mathematical model of intentional perception and action"](http://www.edusoft.ro/brain/index.php/brain/article/view/567):
>When a morphism is between objects of the same category, it is called a homomorphism or hom, and when between objects of different categories it is a heteromorphism or het.
which seems to suggest that the term "homomorphism" is due to the fact that the source and target objects of the relevant morphism are _in the same category_. While that could be the case (considering the prefix _homo-_ means "same"), I also vaguely remember that the term "homomorphism" had already existed before Category Theory was invented, and that in e.g. "group homomorphism" the prefix _homo-_ merely refers to the fact that the source and target objects in question _have the same (or rather ["similar"](https://en.wikipedia.org/wiki/Homomorphism)) structure_. But apparently the term "heteromorphism" only naturally pairs up and contrasts with "homomorphism" in its first sense (i.e. "the same category") but not in its second sense (i.e. "structure-preserving")... :-?