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Hi,
almost by chance I bumped into the following construction. Let be the functor that takes in input a (small) category and returns its twisted arrow category (or, which is the same, the category of elements of the functor ). Let be the nerve functor and let be the functor returning the category of elements of a simplicial set. There is a well-defined natural transformation as follows. For each small category , is the functor acting on the objects as .
My question is: I wonder whether this construction is universal in the sense that is the right Kan extension of along . I don't have a particular reason to believe that, but it type-checks, which is already something. If this is indeed the case I would like to know a bit more what's going on!
Thanks a lot for your patience!