Product (category theory)
Product of an arbitrary family

Instead of two objects, we can start with an arbitrary family of objects indexed by a set I. {\displaystyle I.}

Given a family (X i) i ∈ I {\displaystyle \left(X_{i}\right)_{i\in I}} of objects, a product of the family is an object X {\displaystyle X} equipped with morphisms π i: X → X i, {\displaystyle \pi _{i}:X\to X_{i},} satisfying the following universal property:

The product is denoted ∏ i ∈ I X i. {\displaystyle \prod _{i\in I}X_{i}.} If I = { 1, …, n }, {\displaystyle I=\{1,\ldots,n\},} then it is denoted X 1 × ⋯ × X n {\displaystyle X_{1}\times \cdots \times X_{n}} and the product of morphisms is denoted ⟨ f 1, …, f n ⟩. {\displaystyle \langle f_{1},\ldots,f_{n}\rangle.}
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