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This note has several aims. Firstly, it portrays a non-standard analysis as afunctor, namely a functor * that maps any set A to the set *A of itsnon-standard elements. That functor, from the category of sets to itself, ispostulated to be an equivalence on the full subcategory of finite sets ontoitself and to preserve finite projective limits (equivalently, to preservefinite products and equalizers). Secondly, "Local" non-standard analysis isintroduced as a structure which I call lim-rim, in particular exact lim-rims.The interplay between these, and ultrafilters and ultrapowers, and alsocardinality relations and notions depending on a cardinality such as saturationand what I call "confinement" and "exactness", are investigated.  In particular, one constructs non-standard analyses, with "good" kinds oflim-rims. In these one may say that *A - "the adjunction of all possible limitsfrom A" - plays a role analogous to that of the algebraic closure of a field -"the adjunction of all roots of polynomials". Then in the same spirit as withthe latter, one has uniqueness up to isomorphism, and also universality andhomogeneity, provided one has enough General Continuum Hypothesis. Thecardinality of *A will be something like $2^{2^{|A|}}$, and one has a highdegree of saturation.  Also, one notes that the functor * can be applied again to *A, giving **A,***A, and so forth. In particular, I focus on the two different embeddings of*A into **A and prove some of their properties, with some applications.

Non Standard Analysis as a Functor, as Local, as Iterated