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In this paper we deal with two weaker forms of injectivity which turn out tohave a rich structure behind: separable injectivity and universal separableinjectivity. We show several structural and stability properties of theseclasses of Banach spaces. We provide natural examples of (universally)separably injective spaces, including $\mathcal L_\infty$ ultraproducts builtover countably incomplete ultrafilters, in spite of the fact that theseultraproducts are never injective. We obtain two fundamental characterizationsof universally separably injective spaces: a) A Banach space $E$ is universallyseparably injective if and only if every separable subspace is contained in acopy of $\ell_\infty$ inside $E$. b) A Banach space $E$ is universallyseparably injective if and only if for every separable space $S$ one has$\Ext(\ell_\infty/S, E)=0$. The final Section of the paper focuses on specialproperties of 1-separably injective spaces. Lindenstrauss\ obtained in themiddle sixties a result that can be understood as a proof that, under thecontinuum hypothesis, 1-separably injective spaces are 1-universally separablyinjective; he left open the question in {\sf ZFC}. We construct a consistentexample of a Banach space of type $C(K)$ which is 1-separably injective but not1-universally separably injective. We show that, under the continuumhypothesis, "to be universally separably injective" is not a $3$-spaceproperty, as we wrongly claimed in the paper mentioned in the title.

On separably injective Banach spaces and Corrigendum to "On separably injective Banach spaces" [Adv. Math. 234 (2013) 192--216]