A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of `category with finite products'. To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of `essentially unique' and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which `structure is adjoint to unit', and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-likes, and lax-idempotents are each coreflective among all 2-monads.
Keywords: 2-category, monad, structure, property.
1991 MSC: 18C10,18C15,18D05.
Theory and Applications of Categories, Vol. 3, 1997, No. 9, pp 213-250.