Converts cob(I)alamin to adenosylcobalamin (adenosylcob(III)alamin), a coenzyme for methylmalonyl-CoA mutase, therefore participates in the final step of the vitamin B12 conversion (PubMed:12514191). Generates adenosylcobalamin (AdoCbl) and directly delivers the cofactor to MUT in a transfer that is stimulated by ATP-binding to MMAB and gated by MMAA (By similarity). ATP + cob(I)alamin-[corrinoid adenosyltransferase] = adenosylcob(III)alamin + apo-[corrinoid adenosyltransferase] + triphosphate Homotrimer. Belongs to the Cob(I)alamin adenosyltransferase family.