It is well known that for a functor $F: \mathcal R \to \mathcal M$, where $\mathcal R$ is a Reedy category and $\mathcal M$ is suitably bicomplete, the following decomposition determines the structure around $y \in \mathcal R$: $$ Latch(F)(y) \to F(y) \to Match(F)(y); $$ here we decompose the canonical map between the latching and the matching objects at $y$.

Is there a reference with a written down proof for the same phenomenon occurring in higher categories? Namely, an extension theorem saying that the category of functors from $\mathcal R$ to a bicomplete higher category has the same inductive property, that is, to reconstruct a functor on objects of degree $k$ all we need is to factor the latching-to-matching maps constructed using objects of degree $\leq k-1$?

Thank you in advance.

EDIT: following comments, what interests me is the $(\infty,1)$ case where no presentability is assumed, hence a strictification technique cannot be applied easily.