We present a general method for constructing generic morphisms. Here, ``generic'' may be understood as either being the most complicated one (in a suitable class of morphisms) or having a residual orbit with respect to the natural action of the autormorphism group of its domain. Nevertheless, we shall give a precise and general definition through a natural abstract Banach -- Mazur game.
In certain categories of Banach spaces the above method brings generic non-expansive linear operators. Some of them are universal in the sense that they capture all non-expansive operators with a given range. For example, a universal generic operator exists on the Gurarii space (a result from 2015, joint with J. Garbulinska-Wegrzyn).
As a concrete new application, we show the Kadec -- Pelczynski -- Wojtaszczyk universal space admits a universal generic operator.