Vaughan Pratt wrote in part:
Toby Bartels claims it's obvious, in which case there should be a short construction of a nonconstant function from N^N (as a final coalgebra) to 2 (or to N, the codomain Toby spoke of) in the topos of sheaves on R. Toby, what is it?
An element s of N^N as a final coalgebra of X |-> N x X decomposes into an element s0 of N and an element s+ of N^N. In this notation, the easiest of the desired functions is s |-> s0. Other functions N^N -> N include s |-> s+0, s++0, ...; there are more. This can be done in any topos (actually in more categories than that). If any of these is constant, then N is a terminal object, so if it's an NNO (as it should be to justify the notation N^N) we are in the terminal category (the inconsistent topos). --Toby Bartels