> About Puzzle 97 as stated the answer is infinite, because I can have say two distinct discrete categories of four objects just relabeling objects or identities.
Great! I was waiting to see if anyone would notice this.
Normally when mathematicians ask you to do something like "count the number of groups with four elements", they mean to count _isomorphism classes_ of groups with four elements. Otherwise the answer is usually infinite, and not very interesting, since lots of groups differ only by what names you use for their elements.
That's happening here too. There are infinitely many different (= unequal) categories with four morphisms.
A more interesting interpretation of Puzzle 97, therefore, is to count _isomorphism classes_ of categories with four morphisms. And it looks like that's what people were doing!
The reason I didn't phrase the question more precisely is: 1) it's good for people to discover these issues on their own, and more importantly 2) I couldn't really define "isomorphism of categories" until after we talk about functors.
> Possibly the fun is counting them up to isomorphism or maybe equivalence of categories.
"Equivalence" is interesting too, but a definition of that concept will have to wait until after we talk about natural transformations!
I can say this now, though: if you have two equivalent categories with four morphisms, they must be isomorphic.