## 2012-07-17

### An Odd Lemma

While proving that every monad is an applicative functor, I extracted the following derivation as a lemma:

fmap f ∘ (λh. fmap h x)
≡ { defn. ∘, β-reduction }
λg. fmap f (fmap g x)
≡ { defn. ∘ }
λg. (fmap f ∘ fmap g) x
≡ { Functor }
λg. fmap (f ∘ g) x
≡ { abstract f ∘ g }
λg. (λh. fmap h x) (f ∘ g)
≡ { defn. ∘, η-contraction }
(λh. fmap h x) ∘ (f ∘)

for all f, x. This is the Yoneda Lemma in a special form. The term λh. fmap h x is the natural transformation between an arbitrary functor F and the functor Hom(X, —), where X is fixed by the type of x. Dan Piponi calls it the hardest trivial thing in mathematics. I didn't recognize it immediately (my category-fu is nonexistent), but the striking symmetry made me curious and I started investigating.