(b) Consider the category where the objects are subsets of the set {a,b,c,d}, and where there is a unique morphism from X to Y if X is a subset of Y. Given two subsets, for example {a,b,c} and {b,c,d}, what is their product? What is the name of this binary operation?

@FabricioOlivetti claimed that in the category, the product is the intersection of two sets.

Note that this category is also thin: between sets A and B, there is either no morphism from A to B, if A is not a subset of B, or there is exactly one morphism from A to B, if A is a subset of B.

So a good deal of the structure of my proof for part (a) -- the parts that rely upon thinness -- should carry over.

It would seem kind of repetitive for me to crank out this proof now. Anyone up for getting the ball rolling here?