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In the context of categories enriched in the category of partial order sets, we will work with KZ-reflectivity and Kan-injectivity, notions which, as we shall see, are closely related. We will show that there is an infinite chain of small subcategories of the category Top0, whose Kan-injective hulls are KZ-reflective subcategories of Top0. Moreover, the union of this collection of KZ-reflective subcategories is the full and KZ-reflective subcategory of Top0 which objects are the sober spaces, Sob. By duality, we define the concepts of Kan-projectivity and co-KZ-reflectivity that we study in the category Frm. Here we show that the Kan-projective hulls of the images of the subcategories of An through the contra-variant functor \mathcal{O}:Top0--->Frm, applying each topological space in the frame of the opens, are a chain of KZ-co-reflective subcategories of Frm. But their union is not contained in the subcategory of spatial lattices.
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