I've been studying category theory and on the books, it's never too clear what a morphism really is. Some say that a morphism could be a function, but there are examples of morphisms which are not functions, for example: Morphisms in the category of relations. My guess is that they are property-preserving relations. From Simmons': Introduction to Category Theory, there's a small list of things that are categories:
And from the few ones I know, one could represent morphisms as property-preserving relations. That is, each object in each of the categories can be a set and by taking an appropriate relation, we could define a morphism as being this relation(?). So what is happening?
I'm wrong and seeing morphisms as set of relations would have some kind of problem? (Why?)
They expect us to notice that they are really set of relations?
Another wild guess is that it seems that a category is a concept logically pre-set theory. That is, a category is a concept that can be made without mentioning sets? This seems a little odd because each of the concepts in the list seems to need set theory. I guess the only one we don't need it are categories of categories?