Hi all, this is my first time here... Covid19 and quarantine kind of brought me to this chat :)

Hi @Manuel! Welcome!

@shibai this is far from rude, and far from barging in! i don't know the state of the research on this, but it seems at least in the $n = \infty$ case, there should be an analogue of the gepner--haugseng formalism for enriching in a monoidal (or at least maybe $E_n$ monoidal?) $(\infty,n)$-category. too tired to think through much of this but i'm scared of trying to use un/straightening for $(\infty, n)$-categories; maybe there's something using $\Theta_n$ instead of $\Delta$?

For anyone who's looked into Lurie's "tangent correspondence (HA 7.3.6)" , I have a very basic confusion. In Remark 7.3.6.5 he states the result of computing the fibers of the "fundamental correspondence". I'm assuming this follows from the universal property in the equation right above, but I'm having trouble seeing how to reproduce the result. Any pointers would be appreciated!

Another related question is that in Notation 7.3.6.4. he defines the fundamental correspondence by a universal property (that it represents a certain functor). He doesn't comment on the existence of an object satisfying this universal property - I'm guessing it's obvious but I'm having trouble seeing it.