I pass on to you a curious fact that was mentioned to me by Richard Squire and he credited it to Jim Loveys. Let P and Q be quantifiers. (What is a quantifier? Well as a first approximation it is a map from X--o L to L which is in some obscure (to me) sense polymorphic. Limits (both in categories and posets), integrals and, evidently, univeral and existential quantifiers in logic, where L is the object of truth values. Looking at all the examples, you realize immediately that polymorphism is certainly not functoriality and if some of them give inequalities or morphisms, they do so in different directions (say SUP vs INF or A vs E).) Well, anyway, P and Q are quantifiers. Say that P distributes over Q if there is an equation P(x:X).Q(y:Y).f(x,y) = Q(s:X--o Y).P(x:X).f(x,s(x)) One aspect of polymorphism is that this makes sense. Here f: X x Y --> L and if treat this as a map Y --> (X--o L), then the LHS makes sense. As for the RHS, we are doing the same thing with X x (X--o Y) --> X x Y --> L, where the first map is <id,eval> and the second is f. Here are 2 1/2 examples: If P is universal quantification, Q is existential and L is the object of truth values in a topos, then this is the axiom of choice. If P is sup in a lattice and Q is inf and if X and Y are restricted to finite sets, then this is the statement of distributivity. If P is SUP in a complete lattice and Q is INF, then this is the complete distributivity identity. Now here is the startling fact. In all three cases, the distributivity of P over Q implies that of Q over P! For the lattices, these facts are well-known. For the AC, it is easy to see (at least in a boolean topos) since you can negate and replace f by its complementary relation. (Yes a topos that satisfies AC is boolean, but it does not immediately follow that these two complementary conditions are equivalent, since perhaps only one of them implies AC in the non-boolean case.) This doesn't always hold. For example, in a topos, finite products distribute over (finite) sums, but the converse is certainly false.