Take the ordinary disjoint union, and then add a new _|_ element,
distinct from both existing copies of _|_ (which are still distinct
from each other).
Now why is that not the category-theoretic coproduct?
h . Left = f and h . Right = g both for _|_ and for finite
elements of the types. And it looks universal to me.
Yeah, but there could be more functions from Either X Y to Z than
pairs of functions from X to Z and from Y to Z.
For example, if z :: Z, then you have two functions h1 and h2 such
that h1 . Left = const z and h1 . Right = const z and the same holds
for h2. Namely,
h1 = const z
h2 = either (const z) (const z)
This functions are different : h1 (_|_) = z while h2 (_|_) = (_|_).
And if Either X Y was a category-theoretic coproduct, then the
function from Either X Y to Z would be UNIQUELY determined by it's
restrictions to X and Y.
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