We first unify all notions of partial injectivity appearing in the literature -- (universal) separable injectivity, (universal) א-injectivity -- in the notion of (α, β)-injectivity ((α, β)_λ-injectivity if the parameter λ has to be specified). Then, extend the notion of space of universal disposition to space of universal (α, β)-disposition. Finally, we characterize the 1-complemented subspaces of spaces of universal (α, β)-disposition as precisely the spaces (α, β)₁-injective.

The research of the two first authors has been supported in part by project MTM2016-76958-C2-1-P and Project IB16056 and Ayuda a Grupos GR15152 de la Junta de Extremadura.