Warning: the title refers to the Fixed Point Theorem (aka Diagonal Lemma) in mathematical logic.
So I know that the Fixed Point Theorem is essentially equivalent to the recursion theorem, which is uniform. Does this uniformity imply that the FPT is extensional (i.e. there is a function, ideally computable, that outputs a fixed point given any formula, and the outputs for logically equivalent sentences are also equivalent)? (If you need a base theory, let's say Peano arithmetic or Elementary Arithmetic.)