The product of two categories \(\mathcal{C}\times \mathcal{D}\) may be given as follows:
\(Ob(C\times D)\) are the pairs \((c,d)\) where c is an object of \(\mathcal{C}\) and d is an object of \(\mathcal{D}\).
Morphisms are pairs \((c,d)\xrightarrow{(f,g)}(c',d')\) where \(c \xrightarrow{f}c'\) is a morphism in \(\mathcal{C}\) and \(d \xrightarrow{g}d'\) is a morphism in \(\mathcal{D}\).
Composition is given by composing each entry in the pair separately.