to sign a document that has already been signed by a borrower of money, declaring that, should the borrower be unable to repay that money, the signer will repay the amount due
cosign in American English
(ˈkoʊˌsaɪn; ˈkoʊˈsaɪn)
verb transitive, verb intransitive
1.
to sign (a promissory note) in addition to the maker, thus becoming responsible for the obligation if the maker should default