Let $ \mathbf{A} $ be a category and $ X $ a presheaf on $ \mathbf{A} $ . The category of elements $ \mathbf{E}(X) $ of $ X $ is the category in which: There is a projection functor $ P{\colon}\linebreak[0] \mathbf{E}(X) \to \mathbf{A} $ defined by $ P(A, x) = A $ and $ P(f) = f $ . The following `density theorem’ states that every presheaf is a colimit of representables in a canonical way.