Cosmic magnification on multi-catalogue Herschel submillimetre galaxies

{Submillimetre galaxies (SMGs) are excellent background sources for magnification-bias studies, but the limited sky coverage in the submillimetre (sub-mm) band constrains their statistical power. Beyond H-ATLAS, Herschel produced additional sub-mm catalogues, though not optimised for spatial statistical lensing analyses.} {Our goal is to refine cosmological constraints from SMG magnification bias by exploiting the full sub-mm sky surveyed by Herschel.} {We expanded the SMG sample by incorporating other Herschel catalogues overlapping SDSS spectroscopic lenses. Random catalogues were generated via kernel density estimation to compute cross-correlations, and Markov Chain Monte Carlo methods were applied to infer astrophysical and cosmological parameters for each catalogue and for the combined dataset.} {We report the first detection of magnification bias in SMGs beyond H-ATLAS, reinforcing the robustness of this observable. Individual Herschel catalogues yield reasonable central values for $\Omega_m$ and $\sigma_8$, although with large uncertainties. The combined analysis, dominated by the more powerful H-ATLAS sample, gives results consistent with $\Lambda$CDM: $\Omega_m = 0.30^{+0.05}_{-0.07}$, $\sigma_8 = 0.80 (+/- 0.07)$, and $h < 0.80$, in better agreement with \textit{Planck} 2018 than previous non-tomographic studies.} {SMGs are promising tracers for magnification bias, but the narrow sub-mm coverage remains a major limitation. Wider surveys optimised for lensing would enable cross-correlations on larger scales, yielding tighter cosmological constraints.}