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6
votes
Accepted
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 …
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 …