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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 ?

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The double category of pseudoalgebras, lax and colax morphisms is defined in §5.4 of Grandis and Paré's Multiple categories of generalised quintets. As far as I'm aware, it is the only reference for the construction in the literature. Furthermore, Grandis and Paré show in §5.5 that this constructions extends to a triple category, in which the third kind of morphism is the pseudomorphisms.

Double categories that are symmetric in this sense will tend to be strict double categories, or double bicategories, whereas double categories with functional and relational aspects will tend to be pseudo double categories. Furthermore, in their paper, Grandis and Paré define the notion of double category of quintet type (see §1.4), which captures examples like the double category of pseudo algebras (in particular, this double category is of quintet type).

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    $\begingroup$ I've added the reference to the nLab article. $\endgroup$
    – varkor
    Commented Aug 6 at 19:22
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    $\begingroup$ Thank you much for this reference, this paper looks super cool ! $\endgroup$
    – Axel Osmond
    Commented Aug 6 at 23:32

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