Finite-product-preserving functors,Kan extensions,and strongly-finitary 2-monads

We study those 2-monads on the 2-categoryCat of categories which, as endofunctors, are the left Kan extensions of their restrictions to the sub-2-category of finite discrete categories, describing their algebras syntactically. Showing that endofunctors of this kind are closed under composition involves a lemma on left Kan extensions along a coproduct-preserving functor in the context of cartesian closed categories, which is closely related to an earlier result of Borceux and Day.The first author gratefully acknowledges the support of the Australian Research Council.