A while back, somebody asked me about why automorphisms are always isomorphisms. I bobbled the question a bit. Invertability is always one of those nice things that I take for granted. But he got me wondering. If I have a morphism whose source and target is the same class which is not invertable, what does that mean? I even had to make up the name, semi-automorphism, stealing from the naming of a semigroup, since I could not find a term to describe such a morphism.
What would it mean if a category had such a morphism? Being "structure" preserving, I naturally think of reduction rules in type theory, but is there a more fundamental definition for what they would mean? And, ideally, I'd like to understand this in a way which explains why isomorphisms whose source and target are the same are common enough to earn so much attention in category theory while these non-invertible equivalents do not.
The word you're looking for is "endomorphism", and it is not at all rare for a category to have endomorphisms which are not automorphisms. Consider, for example, the category of vector spaces with morphisms given by linear transformations - not every linear map from a vector space to itself is invertible (i.e., there are square matrices which are not invertible).