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$\DeclareMathOperator\Mnd{Mnd}$I have a bicategory and want to recognize if it's equivalent to $\Mnd(X)$ the bicategory of monads in some other bicategory $X$. Is there a theorem which does this abstractly?

A possible more abstract version of this would be as follows; consider the bicategory of lax functors+lax natural transformations+modifications $_L^L[C,D]$. $\Mnd(X)$ is equivalent to the lax functor bicategory $_L^L[1,X]$. Given $Y$; can we abstractly recognize a bicategory as being of form $_L^L[Y,X]$ for some $X$?

A lower categorical version suggested by this as a first step is; given categories $X$ and $C$ can we recognize $X$ as a functor category $[C,D]$ for some $D$?

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  • $\begingroup$ I'm not sure whether it's relevant to you, but it is possible to characterise the bicategory $\text{EM}(X)$, whose objects and 1-cells are the same as those of $\text{Mnd}(X)$, but whose 2-cells are more general. Depending on your motivation, this might be just as good. $\endgroup$
    – varkor
    Commented Nov 26 at 18:18
  • $\begingroup$ @varkor not directly relevant i think, but i am curious to know about this characterization. The motivation here is getting a handle on possibility of reduction for an extension of your RMnd(j) with a notion of 2-cell to Mnd(K) for some bicategory K in the non-dense case. Is the characterization of EM(X) you have in mind, the one as Eilenberg-Moore completion, or some other one? Timo $\endgroup$
    – Ilk
    Commented Nov 26 at 18:33
  • $\begingroup$ Yes, I mean the completion under Eilenberg–Moore objects, which can be characterised by virtue of being a free completion under a class of weighted limits, e.g. in §4 of Kelly–Schmitt's Notes on enriched categories with colimits of some class. $\endgroup$
    – varkor
    Commented Nov 26 at 18:56
  • $\begingroup$ @varkor Oh I was recently skimning the Lack-Miranda What is the universal property of the 2-category of monads?, but I didn't grok it. That might actually be the answer to my question. $\endgroup$
    – Ilk
    Commented Nov 26 at 19:26
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    $\begingroup$ Lack–Miranda's paper characterises the inclusion $\text{Mnd}(X) \to \text{EM}(X)$, so it is not helpful if you want to characterise $\text{Mnd}(X)$ in isolation. $\endgroup$
    – varkor
    Commented Nov 26 at 19:47

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