So I answered my question that I had already mentioned here earlier (
mathoverflow.net/questions/383061/…) and I wrote a sketch of a proof for the 1-categorical case, and isolated at the end the sort of statement I need in the $(\infty,2)$-categorical case to be happy with the result, so if someone knows where I can find this type of statement, then I would be very grateful ! Or if someone knows who I can ask for this type of thing