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Product (Category Theory)**

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