Catalyzes the covalent attachment of ubiquitin to other proteins. In association with the E3 enzyme BRE1 and LGE1, it plays a role in transcription regulation by catalyzing the monoubiquitination of histone H2B to form H2BK123ub1. H2BK123ub1 gives a specific tag for epigenetic transcriptional activation, elongation by RNA polymerase II, telomeric silencing, and is also a prerequisite for H3K4me and H3K79me formation. In association with the E3 enzyme RAD18, it catalyzes the monoubiquitination of POL30 'Lys-164', involved in postreplication repair of UV-damaged DNA. The RAD6/UBC2-RAD18 complex is also involved in prevention of spontaneous mutations caused by 7,8-dihydro-8-oxoguanine. In association with the E3 enzyme UBR1, is involved in N-end rule-dependent protein degradation. Also involved in sporulation. 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. Forms a heterodimer complexes with the E3 enzymes BRE1, RAD18 and UBR1. Interacts also with UBR2, RTF1, PAF1 and the RNA polymerase II hyperphosphorylated form. The interaction with RNA polymerase II is BRE1- and PAF1-dependent. Up-regulated by UV radiations and during meiosis. The acidic-tail domain of UBC2 mediates interaction with UBR1 and UBR2, and thus is important for polyubiquitination of histones. This domain is also important for sporulation. The N-terminus is blocked. Present with 2770 molecules/cell in log phase SD medium. Belongs to the ubiquitin-conjugating enzyme family.