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Product (Category Theory)**

Given a category $C$ and a collection $(A_{i})_{i \in I}$ of objects in $C$ , a product of the $A_{i}$ is an object $\prod_{i \in I} A_{i}$ of $C$ and for each $i \in I$ an arrow $π_{i} : \prod_{j \in I} A_{j} \to A_{i}$ satisfying the following universal property: for any object $X$ of $C$ with a collection of arrows $(f_{i} : X \to A_{i})_{i \in I}$ , there exists a unique arrow $(\prod_{i \in I} f_{i}) : X \to \prod_{i \in I} A_{i}$ such that $\forall i \in I : π_{i} \circ (\prod_{i \in I} f_{i}) = f_{i}$ . If a product exists, then it is unique up to unique isomorphism (if $(X, (π_{i})_{i \in I})$ and $(Y, (ϕ_{i})_{i \in I})$ are both products of $(A_{i})_{i \in I}$ in $C$ , then there exists a unique arrow $u : X \to Y$ such that $\forall i : π_{i} = ϕ_{i} \circ u$ , and in fact $u$ must be an isomorphism. The categorical product is the prototypical example of a limit.