I'm trying to interpret the join of $(\infty,1)$-category in a more conceptual way. Let me try to explain what I have in mind.

In the classical setting it is almost a triviality to express the join of two categories as a collage along the terminal profunctor: given $\cal C,D$ and the profunctor $\varphi\colon \cal C\to D$ sending all $(C,D)$ to the terminal object in $Set$, the join ${\cal C}\star{\cal D}$ can be written as $ {\cal C}\uplus_\varphi{\cal D}$, where I denote $\uplus_\varphi$ the collage operation.

Now I wonder if, choosing simplicially enriched categories as a model instead of the quasicategorical one, I can describe the join of two $(\infty,1)$-categories $\cal C,D\in{\bf sSet}\text{-}{\bf Cat}$ to be the collage ${\cal C}\uplus_\varphi{\cal D}$ along the terminal $\bf sSet$-profunctor sending any two objects $(C,D)$ to the terminal simplicial set.

Joins of $\infty$-categories are particularly useful to define limits and colimits of diagrams $p\colon K\to\cal C$ to an $(\infty,1)$-category: following T.1.2.13.4 one can define the limit and colimit of $p$ respectively to be the terminal and initial objects in the $(\infty,1)$-categories $\hom_p(\Delta^\bullet\star K,\cal C)$ and $\hom_p(K\star\Delta^\bullet,\cal C)$.

I wonder if this point of view can be generalized to include *weighted* limits/colimits in $(\infty,1)$-categorical language: here is the question.

Consider the simplicial category $\mathfrak C[\Delta^n]$, for any $n$; given a simplicial functor $p\colon\cal K\to C$ between simplicial categories, how can I figure out the join (in the sense described before) $\mathfrak C[\Delta^n]\star \cal K$, and the (simplicial) set of simplicial functors $\hom_p(\mathfrak C[\Delta^n]\star \cal K,\cal C)$ which equal $p$ when restricted to $\cal K$? What is its terminal object, when it exists?

I also wonder if it is possible to encode weighted limit theory using collages along *arbitrary* profunctors, and then translate also this theory in the $(\infty,1)$-categorical setting. Given a generic $\bf sSet$-profunctor $W\colon \mathfrak C[\Delta^n]\to \cal K$, what should the collage $\mathfrak C[\Delta^n]\uplus_W\cal K$ be? What should the terminal object in $\hom_p(\mathfrak C[\Delta^n]\uplus_W\cal K, C)$ be?

Last but not least:

Does the "classical" definition of join between simplicial sets $K\star S$ coincide with the coherent nerve of the simplicial category $\bf K\star S$ obtained as the collage along the terminal $\bf sSet$-profunctor? This would automatically entail that the coherent nerve in $\star$-monoidal, as it is known to be.

**Edit:** It is maybe useful to give an idea about where I began wondering about this: all stemmed from reading this page, which I was trying to complete.