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We present differential cross sections for the $pp \to ppH^{+}H^{-}$ reaction via photon-photon fusion with exact kinematics. We show predictions for $\sqrt{s}$ = 14 TeV (LHC) and at the Future Circular Collider (FCC) for $\sqrt{s}$ = 100 TeV. The integrated cross section for $\sqrt{s}$ = 14~TeV (LHC) is about 0.1~fb and about 0.9~fb at the FCC for $\sqrt{s}$ = 100~TeV when assuming $m_{H^{\pm}} = 150$~GeV. We present distributions in diHiggs boson invariant mass. The results are compared with those obtained within equivalent-photon approximation. We discuss also first calculations of cross section for exclusive diffractive pQCD mechanism with estimated limits on the $g_{hH^{+}H^{-}}$ coupling constant within 2HDM based on the LHC experimental data. The diffractive contribution is much smaller than the $\gamma \gamma$ one. Absorption corrections are calculated differentially for various distributions. In general, they lead to a damping of the cross section. The damping depends on $M_{H^{+}H^{-}}$ invariant mass and on four-momentum transfers squared in the proton line. We discuss a possibility to measure the exclusive production of $H^{\pm}$ bosons.

Production of charged Higgs boson pairs in the $p p \to p p H^+ H^-$ reaction at the LHC and FCC