An $\infty$-categorical localisation functor for diagrams of simplicial sets

Associated to each small category $C$, there is a category of $C$-shaped diagrams of simplicial sets and an $\infty$-category of $NC$-shaped homotopy coherent diagrams of spaces. We present a functor which exhibits the latter as the $\infty$-categorical localisation of the former at the objectwise weak homotopy equivalences. This builds on a Quillen equivalence between the projective and covariant model structures associated to $C$ due to Heuts-Moerdijk, as well as Cisinski's theory of $\infty$-categorical localisations. We use the localisation functor to give simplified proofs that the left (resp. right) homotopy Kan extension of diagrams of simplicial sets presents the $\infty$-categorical left (resp. right) Kan extension of coherent diagrams of spaces.