▁is ▁mon oid ally ▁equivalent ▁to ▁a ▁strict ▁mon oid al ▁category . ▁ ▁Ex amples ▁ ▁Any ▁category ▁with ▁finite ▁products ▁can ▁be ▁regarded ▁as ▁mon oid al ▁with ▁the ▁product ▁as ▁the ▁mon oid al ▁product ▁and ▁the ▁terminal ▁object ▁as ▁the ▁unit . ▁Such ▁a ▁category ▁is ▁sometimes ▁called ▁a ▁cart esian ▁mon oid al ▁category . ▁For ▁example : ▁Set , ▁the ▁category ▁of ▁sets ▁with ▁the ▁Cart esian ▁product , ▁any ▁particular ▁one - element ▁set ▁serving ▁as ▁the ▁unit . ▁Cat , ▁the ▁category ▁of ▁small ▁categories ▁with ▁the ▁product ▁category , ▁where ▁the ▁category ▁with ▁one ▁object ▁and ▁only ▁its ▁identity ▁map ▁is ▁the ▁unit . ▁D ually , ▁any ▁category ▁with ▁finite ▁cop ro duct s ▁is ▁mon oid al ▁with ▁the ▁cop ro duct ▁as ▁the ▁mon oid al ▁product ▁and ▁the ▁initial ▁object ▁as ▁the ▁unit . ▁Such ▁a ▁mon oid al ▁category ▁is ▁called ▁c oc art esian ▁mon oid al ▁R - Mod , ▁the ▁category ▁of ▁modules ▁over ▁a ▁commut ative ▁ring ▁R , ▁is ▁a ▁mon oid al ▁category ▁with ▁the ▁tensor ▁product ▁of ▁modules ▁ ⊗ R ▁serving ▁as ▁the ▁mon oid al ▁product ▁and ▁the ▁ring ▁R ▁( th ought ▁of ▁as ▁a ▁module ▁over ▁itself ) ▁serving ▁as ▁the ▁unit . ▁As ▁special ▁cases ▁one ▁has : ▁ ▁K - V ect , ▁the ▁category ▁of ▁vector ▁spaces ▁over ▁a ▁field ▁K , ▁with ▁the ▁one - dimensional ▁vector ▁space ▁K ▁serving ▁as ▁the ▁unit .