Isometric uniqueness of a complementably universal Banach space for Schauder decompositions

We present an isometric version of the complementably universal Banach space $\mathbb{P}$ with a Schauder decomposition. The space $\mathbb{P}$ is isomorphic to Pe{\l}czy'nski’s space with a universal basis as well as to Kadec’ complementably universal space with the bounded approximation property.