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6 votes
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grothendieck construction for profunctors

[I guess that by $x_*g$ you mean $D(\mathit{id}, g)(x)$ and by $f^*y$ you mean $D(f, \mathit{id})(y)$.] Actually, your second construction is the usual Grothendieck construction for "$\mathbf{Cat}$-v …
Michal R. Przybylek's user avatar
4 votes
Accepted

Enriched Cauchy completions and underlying categories

It works the other way around --- you should have tried to answer your question by yourself before you posted it here :-) Yes, there is an equivalence $[X^{op}, V] \simeq [\overline{X}^{op}, V]$. Yo …
Michal R. Przybylek's user avatar