# Coproduct (category theory)

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This needs fleshing out with things like notation, compared to coproduct and such
Note: see product and coproduct compared for a definition written in parallel with the product definition. This demonstrates how close the concepts are.

## Definition

Given a pair of objects [ilmath]A[/ilmath] and [ilmath]B[/ilmath] in a category [ilmath]\mathscr{C} [/ilmath] a coproduct (of [ilmath]A[/ilmath] and [ilmath]B[/ilmath]) is a:

• Wedge [ilmath]\xymatrix{ A \ar[r]^{i_A} & S & B \ar[l]_{i_B} }[/ilmath] (in [ilmath]\mathscr{C} [/ilmath]) such that:
• for any other wedge [ilmath]\xymatrix{ A \ar[r]^{f_A} & X & B \ar[l]_{f_B} }[/ilmath] in [ilmath]\mathscr{C} [/ilmath]
• there exists a unique arrow [ilmath]S\mathop{\longrightarrow}^mX[/ilmath] (called the mediating arrow) such that the following diagram commutes:
 Diagram of the coproduct of [ilmath]A[/ilmath] and [ilmath]B[/ilmath] [ilmath]\xymatrix{ A \ar[d]_{i_A} \ar[dr]^{f_A} & \\ S \ar[r]^{m} & X \\ B \ar[u]^{i_B} \ar[ur]_{f_B} &}[/ilmath]