The model category of simplicial set valued presheaves on some category C, with the projective model structure, has a universal property: It is the initial model category receiving a functor from C (namely Yoneda embedding followed by the discrete simplicial presheaf functor). That is, for any functor from C into a model category there is a Quillen adjunction from simplicial presheaves on C to that model category "making the triangle commute".

One can think about this in analogy to the Yoneda embedding which makes presheaves on C into the initial cocomplete category (every functor is a colim of representables). Likewise simplicial set valued presheaves can be seen as the the initial hococomplete category (every object is a hocolim of representables). Note that this, unlike the first paragraph, is a statement not about the model category but about the homotopy theory it represents, which is maybe closer to what you are wondering about. Simplicial set valued presheaves with the injective model structure, or cubical set valued presheaves, would have the same property, but not the stricter one from the first paragraph.

This is a result from Dugger's article "Universal homotopy theories", see [his homepage][1], and while you are there by all means take a look at his expository paper "Sheaves and Homotopy Theory".


  [1]: http://pages.uoregon.edu/ddugger/