$\DeclareMathOperator\Mnd{Mnd}$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 category of functors $[C,D]$ for some $D$?