In the context of categories enriched in the category of partial order sets, we will work with KZreflectivity and Kaninjectivity, notions which, as we shall see, are closely related. We will show that there is an infinite chain of small subcategories of the category Top_{0}, whose Kaninjective hulls are KZreflective subcategories of Top_{0}. Moreover, the union of this collection of KZreflective subcategories is the full and KZreflective subcategory of Top_{0} which objects are the sober spaces, Sob. By duality, we define the concepts of Kanprojectivity and coKZreflectivity that we study in the category Frm. Here we show that the Kanprojective hulls of the images of the subcategories of A_{n} through the contravariant functor \mathcal{O}:Top_{0}>Frm, applying each topological space in the frame of the opens, are a chain of KZcoreflective subcategories of Frm. But their union is not contained in the subcategory of spatial lattices.
