I'm currently working with an analogue of the Grothendieck construction for enriched categories:

Given a distributor a.k.a. $\mathbb V$-functor $D:X^\mathrm{op}\otimes Y\to \mathbb V$ there is a structure of a $\mathbb V$-category on the set $$ \coprod_{X\times Y}\mathbb V(I, D(x,y))=\{\,(x,\phi,y)\,\,|\,\, \phi:I\to D(x,y),x\in X,y\in Y\,\} $$ with $[(x,\phi,y),(x',\phi',y')]$ given by the pullback of the cospan $$ [x,x']\to [x,x']\otimes D(x',y') \to D(x,y') \leftarrow D(x,y)\otimes [y,y'] \leftarrow [y,y']. $$ This construction directly generalises the Grothendieck construction for $\mathrm{Set}$-valued distributors between ordinary categories.

**Question:** What literature does allready exist on this subject?

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