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.