Let $T$ be some object of $C$, and let $a\colon X \to T$ and $b\colon Y \to T$ be morphisms of $C$. Then, by definition of coproduct, there exists a unique morphism $[a,b]\colon X \sqcup Y \to T$ such that the obvious diagrams commute.

This $[a,b]$ is the copairing of $a$ and $b$.

Notation

When convenient, it is nice to write it vertically; all of the following are seen:

The vertical notation can be combined with pairing to create a matrix calculus for morphisms from a coproduct to a product. This works best when products and coproducts are the same, as described at matrix calculus.

Examples

One often sees a function defined by cases as follows:

Such a definition is valid in general iff the domain of $f$ is the (internal) disjoint union of its subsets$\{x : \phi(x)\}$ and $\{x : \psi(x)\}$. In that case, let $X$ be the first subset, let $Y$ be the second, and let $T$ be the target of $f$. Then we have $X \sqcup Y$ as the domain of $f$, and $f$ itself is the copairing of $g$ and $h$, so that

$f = \left[{g \atop h}\right]\,.$

Last revised on March 7, 2023 at 04:15:35.
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