Definition:Canonical Mapping from Coproduct to Product

Definition
Let $C$ be a category with zero morphisms.

Let $a, b$ be objects of $C$.

Assume they have a coproduct $(a \sqcup b, i_1, i_2)$ and a product $(a \times b, p_1, p_2)$.

Definition 1
Let:
 * $j_1 : a \to a \times b$ be the unique morphism such that:
 * $p_1 \circ j_1 = 1 : a \to a$
 * $p_2 \circ j_1 = 0 : a \to b$
 * $j_2 : b \to a \times b$ be the unique morphism such that:
 * $p_1 \circ j_2 = 0 : b \to a$
 * $p_2 \circ j_2 = 1 : b \to b$

The canonical mapping from $a \sqcup b$ to $a \times b$ is the unique morphism $r : a \sqcup b \to a \times b$ such that:
 * $r \circ i_1 = j_1$
 * $r \circ i_2 = j_2$

Definition 2
Let:
 * $q_1 : a \sqcup b \to a$ be the unique morphism such that:
 * $q_1 \circ j_1 = 1 : a \to a$
 * $q_1 \circ j_2 = 0 : b \to a$
 * $q_2 : a \sqcup b \to b$ be the unique morphism such that:
 * $q_2 \circ j_1 = 0 : a \to a$
 * $q_2 \circ j_2 = 1 : b \to a$

The canonical mapping from $a \sqcup b$ to $a \times b$ is the unique morphism $r : a \sqcup b \to a \times b$ such that:
 * $p_1 \circ r = q_1$
 * $p_2 \circ r = q_2$

Also see

 * Equivalence of Definitions of Canonical Mapping from Coproduct to Product
 * Definition:Natural Transformation from Coproduct Functo to Product Functor