Factorship is a relatively uncommon term that refers to the state or condition of being a factor, which is a commercial agent who is entrusted with the task of negotiating the sale of goods or services on behalf of a principal. Phonetically, the word is transcribed as /ˈfæktərʃɪp/. The first syllable "fact-" is pronounced with a short 'a' as in 'cat', followed by the schwa sound in the second syllable "-tər," and ending with the stressed syllable "-ship," pronounced with a short 'i' as in 'ship'.
The business of a factor.
Etymological and pronouncing dictionary of the English language. By Stormonth, James, Phelp, P. H. Published 1874.