Left Kan extensions that are algebraic over colax-idempotent 2-monads

Using the language of double categories we generalise a classical result on finite-product-preserving left Kan extensions, by Ad'amek and Rosick'y, to one on left Kan extensions that preserve algebraic structures defined by suitable' colax-idempotent 2-monads, as well as obtain two related results. To be precise, by suitable’ 2-monads here we mean ones that extend to normal lax double monads. In an appendix we consider induced algebra structures on presheaf objects.