We complete the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We have established some relations with forcing axioms and with the existence of certain regular forcing embeddings and projections, and also point out connections with Precipitousness. We give an rough overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory. In this talk we finish explaining the mechanism of absorbing extenders in the core model, and lifting iterability from countable models to models of large cardinality.