Cotupling

Cotupling is a construction used in category theory and functional programming to combine two maps with a common codomain into a single map from a coproduct (sum type) to that codomain. It is the dual notion to copairing: given a coproduct A+B with canonical injections i1: A -> A+B and i2: B -> A+B, and given maps f: A -> X and g: B -> X, there exists a unique morphism h: A+B -> X such that h ∘ i1 = f and h ∘ i2 = g. This morphism h is called the cotuple or copairing of f and g, often denoted [f,g].

In programming terms, cotupling corresponds to defining a function from a sum type to a common target by applying the appropriate function to each case. For example, with a sum type equivalent to Either A B and a target X, cotupling is a function that maps Left a to f a and Right b to g b. In Haskell-like notation, cotuple f g (Left a) = f a; cotuple f g (Right b) = g b, or equivalently using a case/either expression: cotuple f g x = either f g x.

Cotupling is thus the pattern of converting two independent functions into a single function that handles the combined input type. It is closely related to the universal property of coproducts and is often described as copairing. The concept is commonly contrasted with currying/uncurrying and with tupling in the dual setting of products, where a single map from a product type A×B to X is obtained from maps A -> X and B -> X.