1-complemented subspaces of Banach spaces of universal disposition

We first unify all notions of partial injectivity appearing in the literature ---(universal) separable injectivity, (universal) $\aleph$-injectivity --- in the notion of $(\alpha, \beta)$-injectivity ($(\alpha, \beta)_\lambda$-injectivity if the parameter $\lambda$ has to be specified). Then, extend the notion of space of universal disposition to space of universal $(\alpha, \beta)$-disposition. Finally, we characterize the $1$-complemented subspaces of spaces of universal $(\alpha, \beta)$-disposition as precisely the spaces $(\alpha, \beta)_1$-injective.