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Jul 21, 2018 at 21:41 comment added Noam Zeilberger OK, sorry I didn't ask if you knew this reference (but I think you should indeed revise the question to mention it, since it contains an explicit proof of the result when $\mathcal{K} = \mathbf{Cat}$). There is also a statement of the general 2-categorical case in On monads and warpings, but without proof.
Jul 21, 2018 at 20:35 comment added fosco To put it shortly, a formal justification of the validity of the axioms (in terms of, say, universal properties of left extensions) is buried in lots of daunting equations... if all else fails I can simply point to the paper, but I don't want to cheat.
Jul 21, 2018 at 20:33 comment added fosco Yes, I know that source (I should have pointed to the paper in the OP). It is in fact precisely what I am trying to generalize, but to no avail until now.
Jul 21, 2018 at 16:23 comment added Noam Zeilberger (I wouldn't say it is conceptual, but it is very explicit and can probably be adapted to any 2-category.)
Jul 21, 2018 at 16:17 history answered Noam Zeilberger CC BY-SA 4.0