We study two categories of cellular automata. First, for any group G, we consider the category \(\mathcal{C}\mathcal{A}(G)\) whose objects are configuration spaces of the form \(A^G\) , where A is a set, and whose morphisms are cellular automata of the form \(\tau : A_1^G \rightarrow A_2^G\) . We prove that the categorical product of two configuration spaces \(A_1^G\) and \(A_2^G\) in \(\mathcal{C}\mathcal{A}(G)\) is the configuration space \((A_1 \times A_2)^G\) . Then, we consider the category of generalized cellular automata \(\mathcal {GCA}\) , whose objects are configuration spaces of the form \(A^G\) , where A is a set and G is a group, and whose morphisms are \(\phi \) -cellular automata of the form \(\mathcal {T} : A_1^{G_1} \rightarrow A_2^{G_2}\) , where \(\phi : G_2 \rightarrow G_1\) is a group homomorphism. We prove that a categorical weak product of two configuration spaces \(A_1^{G_1}\) and \(A_2^{G_2}\) in \(\mathcal {GCA}\) is the configuration space \((A_1 \times A_2)^{G_1 *G_2}\) , where \(G_1 *G_2\) is the free product of \(G_1\) and \(G_2\) . The previous results allow us to naturally define the product of two cellular automata in \(\mathcal{C}\mathcal{A}(G)\) and a weak product of two generalized cellular automata in \(\mathcal {GCA}\) .