As explained on this nlab page, for a 2-monad there is a double category of (strict) algebras, horizontal morphisms are lax morphisms and vertical morphisms are colax morphisms of algebras.

However it seems difficult to find a reference for this beside the nlab page: does anyone know a source with complementary information on this ?

More generally, are there sources that investigate this kind of more symmetric double categories where the vertical and horizontal morphisms look like dual classes of morphisms, as opposed to those akin to equipments where one class of morphisms is a relational generalization of the other one ?