I also asked this question on MathOverflow, in case anyone is interested to follow. https://mathoverflow.net/questions/324506/is-this-lifting-of-set-into-cat-a

I also asked this question on MathOverflow, in case anyone is interested to follow. https://mathoverflow.net/questions/324506/is-this-lifting-of-set-into-cat-a-valid-construction https://mathoverflow.net/questions/324506/is-this-lifting-of....