Abstract and concrete tangent modules on Lipschitz differentiability spaces

We construct an isometric embedding from Gigli's abstract tangent module into the concrete of a space admitting (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when is isomorphism. Together with arguments recent article by Bate--Kangasniemi--Orponen, this equivalence used to show that ${\rm Lip}-{\rm lip}$ -type condition lip} f\le C|Df|$ implies existence moreover self-improves f =|Df|$. We also provide direct proof result Gigli second author that, for strongly rectifiable decomposition, admits so-called Gromov--Hausdorff module, without any priori reflexivity assumptions.