This comment makes a fair point that people might be missing at first glace. It’s true, as IainM said, that many of the examples will be of the form “Remember that concept from Abstract Algebra of a free group generated by a set? Remember also the concept of the underlying set of elements in a group? Turns out that those concepts are an example of an adjoint pair of functors.” If you don’t have any recollection of the motivating concepts to draw upon, those examples won’t be of much help to you.
However, any category theory text will give you some example without assuming any prior familiarity. For example, preorders and posets, groups (defined as single-object categories in which all morphisms are invertible) and groupoids, plus the constructions that can be built out of given categories, such as comma categories, functor categories, and products of categories.
That said, if you don’t have some prior familiarity with Set Theory and the set-theoretical definition of functions, you will have a rough time. Most authors I’ve seen want to be able to invoke Set as an example of a category without having to explain to you what a set is or what functions between sets are. (Though, Goldblatt is an exception. He tries to teach the reader set theory before moving on to the category theory.)
This comment makes a fair point that people might be missing at first glace. It’s true, as IainM said, that many of the examples will be of the form “Remember that concept from Abstract Algebra of a free group generated by a set? Remember also the concept of the underlying set of elements in a group? Turns out that those concepts are an example of an adjoint pair of functors.” If you don’t have any recollection of the motivating concepts to draw upon, those examples won’t be of much help to you.
However, any category theory text will give you some example without assuming any prior familiarity. For example, preorders and posets, groups (defined as single-object categories in which all morphisms are invertible) and groupoids, plus the constructions that can be built out of given categories, such as comma categories, functor categories, and products of categories.
That said, if you don’t have some prior familiarity with Set Theory and the set-theoretical definition of functions, you will have a rough time. Most authors I’ve seen want to be able to invoke Set as an example of a category without having to explain to you what a set is or what functions between sets are. (Though, Goldblatt is an exception. He tries to teach the reader set theory before moving on to the category theory.)