Accepts ubiquitin from the E1 complex and catalyzes its covalent attachment to other proteins. In vitro catalyzes 'Lys-11'- and 'Lys-48'-linked polyubiquitination. Acts as an essential factor of the anaphase promoting complex/cyclosome (APC/C), a cell cycle-regulated ubiquitin ligase that controls progression through mitosis. Acts by initiating 'Lys-11'-linked polyubiquitin chains on APC/C substrates, leading to the degradation of APC/C substrates by the proteasome and promoting mitotic exit. 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. S-ubiquitinyl-[E1 ubiquitin-activating enzyme]-L-cysteine + [acceptor protein]-L-lysine = [E1 ubiquitin-activating enzyme]-L-cysteine + N(6)-monoubiquitinyl-[acceptor protein]-L-lysine. Protein modification; protein ubiquitination. Component of the APC/C complex, composed of at least 14 distinct subunits that assemble into a complex of at least 19 chains with a combined molecular mass of around 1.2 MDa. Within this complex, directly interacts with ANAPC2. Autoubiquitinated by the APC/C complex, leading to its degradation by the proteasome. Its degradation plays a central role in APC/C regulation, allowing cyclin-A accumulation before S phase entry. APC/C substrates inhibit the autoubiquitination of UBE2C/UBCH10 but not its E2 function, hence APC/C remaining active until its substrates have been destroyed. Belongs to the ubiquitin-conjugating enzyme family.