This talked was based on old work which I recently wrote up as Limits for lax morphisms.
It concerns limits in the 2-category of strict algebras and lax morphisms for a 2-monad. This includes both the 2-category of monoidal categories and monoidal functors as well as the 2-category of monoidal categories and opomonoidal functors. It also includes the 2-category of categories with a particular class of colimits and {\em arbitrary} functors between them.
I showed how such 2-categories admit oplax limits, including the oplax limit of an arrow, products, and cotensor products; as well as co-Eilenberg-Moore objects (for comonads), and certain inserters, equifiers, and comma objects. (The inserters, equifiers, and comma objects in question are those for which a particular one of the arrows is not just a lax morphism but a pseudo one.)
I described an alternative possible approach to the theory involving the horizontal double limits of Grandis and Paré.