Product logic is, together with Łukasiewicz logic and Gödel logic, one of the fundamental logics in Hájek’s framework of fuzzy logics arising from a continuos t-norm. In algebraic terms, this means that product algebras are one of the most relevant subvarieties of BL-algebras. From the algebraic perspective, representations of product algebras have mostly highlighted their connection with cancellative hoops; however, not much is known about the relation between product algebras and the variety of residuated lattices that constitutes their 0-free subreducts: product hoops. This contribution focuses on two main results: 1) we prove that product hoops coincide with the class of maximal filters of product algebras, seen as residuated lattices; 2) we show a construction that given any product hoop H obtains a product algebra freely generated by H; in terms of the corresponding algebraic categories, we exhibit the free functor, i.e. the left adjoint to the forgetful functor from product algebras to product hoops.