Homotopy theory of small diagrams over large categories

Let 𝒟 be a large category which is cocomplete. We construct a model structure (in the sense of Quillen) on the category of small functors from 𝒟 to simplicial sets. As an application we construct homotopy localization functors on the category of simplicial sets which satisfy a stronger universal property than the customary homotopy localization functors do.