Existence of categorical products in the category of generalized Set-cellular automata (GCA_Set)

Determine whether binary products exist in the category GCA_Set of generalized Set-cellular automata; specifically, for any sets A and B and groups G and H, ascertain whether there exists an object P in GCA_Set together with generalized Set-cellular automata p_A: P → A^G and p_B: P → B^H such that for every object C^K in GCA_Set and pair of generalized Set-cellular automata α: C^K → A^G and β: C^K → B^H, there exists a unique generalized Set-cellular automaton u: C^K → P with p_A ∘ u = α and p_B ∘ u = β.

Background

The paper introduces the category CA_C(G) of C-cellular automata over a fixed group G and proves that CA_C(G) is closed under finite products. It then generalizes to the category GCA_C of generalized C-cellular automata, where morphisms can connect configuration objects over different groups via group homomorphisms, and shows the existence of a finite weak product constructed using the product in C and the free product of the underlying groups.

However, the authors show that the weak product in GCA_C need not be a true product because the factorization of generalized cellular automata is not unique in general, and in the concrete case C = Set, constant generalized cellular automata never have unique factorizations. Consequently, while a weak product exists, uniqueness fails in GCA_Set, prompting the explicit question of whether any product (satisfying the full universal property) exists in GCA_Set.