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In this study, we developed a new proton detector system called CATCH, which was designed for a scattering experiment involving a $\Sigma$ hyperon and a proton (J-PARC E40). CATCH is a cylindrical detector system covering an inner target that can be used to measure the trajectory and energy of a proton emitted from the target for the kinematic identification of a $\Sigma$p scattering event. It comprises a cylindrical fiber tracker (CFT), a bismuth germanate (BGO) calorimeter, and a plastic scintillator hodoscope (PiID), which are coaxially arranged from the inner to the outer sides. The CFT is a tracking detector consisting of 5,000 scintillation fibers, and it has two types of cylindrical layers in which the fibers are arranged in straight and spiral configurations. 24 BGO crystals are placed around the CFT to measure the kinetic energy of the recoil proton. The PiID is used to determine whether the recoil proton is stopped in the BGO calorimeter. We performed proton-proton (pp) and proton-carbon (pC) scattering experiments using an 80 MeV proton beam to evaluate the performance of CATCH. The total energy resolution for the recoil proton was 2.8 MeV in $\sigma$ for the entire angular region after the energy calibrations of the BGO calorimeter and the CFT. The angular resolution of the CFT was 1.27 degrees in $\sigma$ for the proton, and the time resolution was more than 1.8 ns in $\sigma$. We also developed an analysis method for deriving the cross section of the pp scattering using CATCH. The obtained relative differential cross section for the pp elastic scattering was consistent with that obtained by reliable partial wave analysis, and the systematic error was maintained at below 10%. These performance results satisfy our requirements for a reliable detection system for the $\Sigma$p scattering experiment conducted at J-PARC .