I don’t understand how the homology groups in regards to the derived category of sheafs on a space X is connected to the cohomology of a sheaf which is calculated with the images/kernels after applying the global sections functor. As we have a quasi-isomorphism between sheaf complexes if the homology groups (which are calculated with images/kernels of maps of sheafs - not global sections) are isomorphic.
What am I missing?