Catalyzes the covalent attachment of ubiquitin to other proteins. Mediates the selective degradation of short-lived and abnormal proteins. Mediates ubiquitination of PEX5. S-ubiquitinyl-[E1 ubiquitin-activating enzyme]-L-cysteine + [E2 ubiquitin-conjugating enzyme]-L-cysteine = [E1 ubiquitin-activating enzyme]-L-cysteine + S-ubiquitinyl-[E2 ubiquitin-conjugating enzyme]-L-cysteine. Protein modification; protein ubiquitination. Belongs to the ubiquitin-conjugating enzyme family.