# Product (category theory)

(Redirected from Categorical product)
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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 a coproduct 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 product (of [ilmath]A[/ilmath] and [ilmath]B[/ilmath]) is a:

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