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definition. product category [leinster2016basic, 1.1.11] [tt-0048]
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    Given categories \({\cal C}\) and \({\cal D}\), there is a product category, denoted \({\cal C} \times {\cal D}\), in which

    • an object is a pair \((X, Y)\)
    • an arrow \((X, Y) \to \left (X', Y'\right )\) is a pair \((f, g)\)
    • the composition is given by \[(f_1, g_1) \mathbin {\bullet } (f_2, g_2) = (f_1 \mathbin {\bullet } f_2, g_1 \mathbin {\bullet } g_2)\]
    • the identity on \((X, Y)\), denoted \(\mathit {1}_{(X, Y)}\) is \((\mathit {1}_X, \mathit {1}_Y)\)
    where \(X \in {\cal C}\), \(Y \in {\cal D}\), \(f: X \to X' \in {\cal C}\), and \(g: Y \to Y' \in {\cal D}\).