Sure, take the category of simplicial sets, which is Hom(Δop, Set). We don't usually think of it as a way of studying the category Δ! There are many other examples along these lines in homotopy theory.
Edit: To elaborate slightly more on what I had in mind with the last comment, take a look at my answer here" where I describe a presentation of the category of monoids, i.e., a way to embed it as a reflective subcategory of a presheaf category (in fact, the category of simplicial sets). This point of view is more commonly encountered in homotopy theory, because to get a good, non-strict notion of, say, topological monoid, one cannot simply write down operations with relations that are required to hold on the nose. This kind of presentation as objects of a presheaf category which send some diagrams to homotopy limit diagrams is one way to resolve the issue.