information:classification
Examples

In the category of sets, the product (in the category theoretic sense) is the Cartesian product. Given a family of sets X i {\displaystyle X_{i}} the product is defined as

with the canonical projections

Given any set Y {\displaystyle Y} with a family of functions f i: Y → X i, {\displaystyle f_{i}:Y\to X_{i},} the universal arrow f: Y → ∏ i ∈ I X i {\displaystyle f:Y\to \prod _{i\in I}X_{i}} is defined by f (y):= (f i (y)) i ∈ I. {\displaystyle f(y):=\left(f_{i}(y)\right)_{i\in I}.}

Other examples: