How is the weight $J(-)$ determined in this characterization of the category of element of a copresheaf as a coend? http://ncatlab.org/nlab/show/category+of+elements#properties_11
The functor el(-) seems to admit a right adjoint given by $D \to K(D)$ : $c\mapsto Ob([Jc, D])$, which would imply cocontinuity; am I right in deducing this from a purely formal-nonsense argument?
I'm trying to unravel the coend definition in the simple example of the action groupoid of a G-set, to see if I'm able to find what $J$ has to be at least in that case. Maybe this is linked to the presence of an adjoint for the functor $X\mapsto X//G$ from G-sets to groupoids. Does anybody know a reference for that?
Thanks.