Editing Openai/693343d7-a38c-8012-a67c-11cbed4c0fd9 (section)

===== We need that O is parametric in the sense that there is a canonical iso: =====

<syntaxhighlight>O (R -> A) ≅ (R -> O A)

</syntaxhighlight>

for all A — i.e., O commutes with products/exponentials for constant R. This holds if O is defined polynomially / functorially with no dependence on R, e.g., Tree built from A and finite lists:

For Tree(A):

<syntaxhighlight>Tree (R -> A)  ≅  R -> Tree A

</syntaxhighlight>

? Not strictly isomorphic in general, but if Tree is built from A in strictly covariant positions, function-space commutation requires parametricity/uniformity. However there is a natural map:

<syntaxhighlight>φ_A : (R -> Tree A) -> Tree (R -> A)

</syntaxhighlight>

constructed by mapping a function F : R -> Tree A to a Tree (R -> A) that at each leaf yields the function r ↦ value at leaf — this requires exchanging quantifiers and may not be an iso unless Tree is finitary and R is discrete.