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Add LoadPath "..".
Require Import Homotopy.
Lemma adjunction_product_hom (X Y Z : Type) : (X -> (Y -> Z)) ≃> (X * Y -> Z).
Proof.
set (left_to_right := fun (f : X -> (Y -> Z)) => fun (xy : X * Y) => let (x, y) := xy in f x y).
set (right_to_left := fun (f : X * Y -> Z) => fun x => fun y => f (x, y)).
exists left_to_right.
apply hequiv_is_equiv with (g := right_to_left).
intro y.
apply funext; intro t.
destruct t.
apply idpath.
intro x.
apply funext; intro t.
apply funext; intro t'.
apply idpath.
Defined.
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