The Hopf algebras \(U_q ({\mathfrak g})\) and \(U_q (\widehat {\mathfrak g})\) are to be understood as in the previous paper (see the review above). Regarding \(U_q ({\mathfrak g})\) as a subalgebra of \(U_q (\widehat {\mathfrak g})\) the question arises whether a representation of \(U_q ({\mathfrak g})\) extends to a representation of \(U_q (\widehat {\mathfrak g})\), or more general, how to enlarge the representation space to get some extension. An affinization of a finite-dimensional irreducible representation \(V\) of \(U_q ({\mathfrak g})\) is by definition an irreducible representation \(\widehat {V}\) of \(U_q (\widehat {\mathfrak g})\) containing \(V\) as a \(U_q ({\mathfrak g})\)-subrepresentation with multiplicity one and such that all other \(U_q ({\mathfrak g})\)-subrepresentations of \(\widehat {V}\) are strictly smaller than \(V\). It is proved (Proposition 3.5) that the set of affinizations is finite up to isomorphisms and defining a partial ordering (Definition 3.8) the existence of a minimal affinization is evident. In the rank 2 case, the question of uniqueness of the minimal affinization is answered (Theorem 5.1), and the structure of the minimal affinization as a \(U_q ({\mathfrak g})\)-module is given in the case that \({\mathfrak g}\) is of type \(A_2\) or \(C_2\) (Theorem 6.1).