  We establish the existence of the long-debated $f_0(1370)$ resonance in the
dispersive analyses of meson-meson scattering data. For this, we present a
novel approach using forward dispersion relations, valid for generic inelastic
resonances. We find its pole at $(1245\pm 40)- i\,(300^{+30}_{-70})$ MeV in
$\pi\pi$ scattering. We also provide the couplings as well as further checks
extrapolating partial-wave dispersion relations or with other continuation
methods. A pole at $(1380^{+70}_{-60})-i\,(220^{+80}_{-70})$ MeV also appears
in the $\pi\pi\to K\bar K$ data analysis with partial-wave dispersion
relations. Despite settling its existence, our model-independent dispersive and
analytic methods still show a lingering tension between pole parameters from
the $\pi\pi$ and $K\bar K$ channels that should be attributed to data.
