We perform the combined analysis of the double Higgs production via gluon fusion in the bb̅γγ and bb̅τ^+τ^- decay channels at the High-Luminosity LHC (HL-LHC). To validate our analysis, we reproduce the ATLAS result of the bb̅γγ process including all contributions from fakes. For the bb̅τ^+τ^- decay channel, we perform the similar analysis to the CMS one. As an improvement, we also perform the multivariate analysis employing the boosted decision tree algorithm. Then, we derive 68 field theory

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... FT) approach for various analyses. We find that the bb̅τ^+τ^- process outperforms the bb̅γγ for the measurement of energy-growing operators, while adding the bb̅τ^+τ^- process is least beneficial for improving the precision of the Higgs self-coupling (mainly set by the bb̅γγ process). We illustrate that the double Higgs production alone can be comparable to the single Higgs process in constraining the modification of the top Yukawa coupling in the positive direction. Focusing on the Higgs self-coupling as a special interest, we derive the precision as a function of various improvable parameters such as tag and mistag rates of tau leptons, heavy flavor jets, photon identification, diphoton mass resolution, and jet energy resolution to take into account future phenomenological studies. As an optimistic benchmark scenario, we illustrate that the 68 self-coupling, λ_3/λ_3^SM, at the HL-LHC can reach [0.2, 2.3] and [-0.1, 3.5] ∪ [4.0, 6.5], respectively, where the correlation among the EFT coefficients is taken into account.