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Asad Saeeduddin (Mar 06 2021 at 04:05):I learnt today from the "Iterated Monoidal Categories" paper that a 2-category of strict-monoidal categories and lax monoidal functors between them has a product (derived from the product in Cat). Does LaxMonCat have an analogous product?
Fawzi Hreiki (Mar 06 2021 at 09:26):The coherence theorem for monoidal categories says that the inclusion of strict monoidal categories into monoidal categories has a left adjoint so if this product is the cartesian product then yes.
Mike Shulman (Mar 06 2021 at 15:07):Yes. In fact, the result you mention has a generalization: the 2-category of strict algebras and lax morphisms for any 2-monad inherits products from products in its base category. This is Corollary 4.9 of Lack's Limits for lax morphisms. This implies the question you ask about, since there is a 2-monad whose strict algebras are lax monoidal categories (and another one whose strict algebras are the usual kind of "pseudo" monoidal categories).