Abstract: Background: The evolution of shell structure around doubly magic exotic nuclei is of great interest in nuclear physics and astrophysics. In the 'southwest' region of Ni-78, the development of deformation might trigger a major shift in our understanding of explosive nucleosynthesis. To this end, new spectroscopic information on key close-lying nuclei is very valuable. Purpose: We intend to measure the isomeric and beta decay of Co-75, with one-proton and two-neutron holes relative to Ni-78, to access new nuclear structure information in Co-75 and its beta-decay daughters Ni-75 and Ni-74. Methods: The nucleus Co-75 is produced in relativistic in-flight fission reactions of U-238 at the Radioactive Ion Beam Factory in the RIKEN Nishina Center. Its isomeric and f decay are studied exploiting the BigRIPS and EURICA setups. Results: We obtain partial beta-decay spectra for Ni-75 and Ni-74, and report a new isomeric transition in Co-75. The energy [E-gamma = 1914(2) keV] and half-life [t(1/2) = 13(6) μs] of the delayed gamma ray lend support for the existence of aJ(pi) = (1/2(-)) isomeric state at 1914(2) keV. A comparison with PFSDG-U shell-model calculations provides a good account for the observed states in Ni-75, but the first calculated 1/2(-) level in Co-75, a prolate K = 1/2 state, is predicted about 1 MeV below the observed (1/2(-)) level. Conclusions: The spherical-like structure of the lowest-lying excited states in Ni-75 is proved. In the case of Co-75, the results suggest that the dominance of the spherical configurations over the deformed ones might be stronger than expected below Ni-78. Further experimental efforts to discern the nature of the J(pi) = (1/2(-)) isomer are necessary.