Sure, take the category of simplicial sets, which is Hom(&Delta;<sup>op</sup>, Set).  We don't usually think of it as a way of studying the category &Delta;!  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 <a href="http://mathoverflow.net/questions/6376/why-forgetful-functors-usually-have-left-adjoint/6533#6533">here"</a> 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.