A note on universal operators between separable Banach spaces

We compare two types of universal operators constructed relatively recently by Cabello S'anchez, and the authors. The first operator $\mathrm{\Omega }$ acts on the Gurarii space, while the second one $P_S$ has values in a fixed separable Banach space $S$. We show that if $S$ is the Gurarii space, then both operators are isometric. We also prove that, for a fixed space $S$, the operator $P_S$ is isometrically unique. Finally, we show that $\mathrm{\Omega }$ is generic in the sense of a natural infinite game.