Let B be a k-bialgebra (associative, coassociative, unitary and counitary) and g an indeterminate. Denote B*k[g] the free product of the k-algebras B and k[d]. Declare g to be group-like, so that B*k[d] is again a bialgebra, and kg a subcoalgebra (hence a (left and right) comodule). The question is:

Is the category of B*k[g]-comodules generated (under tensor products, sums and summands) by kg and B-comodules? Is there a reference to point out?