On any reasonable definition of functions, neither the category of sets nor the category of small categories is cartesian closed in New Foundations (NF). The latter category is sometimes proposed as a foundation for category theory since it is among its own objects. Our result shows it is a poor one.

In NF, as in other set theories, a "function" f from a set A to a set B is defined to be a set f of ordered pairs 〈x, y〉 with x in A and y in B, such that (a) if 〈x, y〉 ∈ f and 〈x, y′〉 ∈ f then y = y′, and (b) for every x in A there is some y in B with 〈x, y〉 ∈ f. But in NF different definitions of ordered pairs give significantly different functions. I say a reasonable definition must give:

1. The formula z = 〈x, y〉 is stratifiable.

2. For every set S there is a set {〈x, x〉 ∣ x ∈ S}.

3. If f is a function from A to B, and g one from B to C, there is a set {〈x, z〉∣(∃y)〈x, y〉∈ f & 〈y, z〉∈ g}.

Principles 2 and 3 are needed for identity functions and composites. By principle 1, any sets A and B have a set A × B of all ordered pairs 〈x, y〉 with x in A and y in B, but it does not follow that functions exist making A × B a categorical product of A and B.