▁every ▁diagram ▁comm utes ▁in ▁a ▁pre order . ▁ ▁Note ▁that ▁if ▁C ▁is ▁a ▁partial ▁order , ▁the ▁above ▁description ▁is ▁simplified ▁even ▁more , ▁because ▁the ▁associ ativity ▁and ▁un ital ity ▁is omorph isms ▁becomes ▁equal ities . ▁Another ▁simpl ification ▁occurs ▁if ▁we ▁assume ▁that ▁the ▁set ▁of ▁objects ▁is ▁the ▁free ▁mon oid ▁on ▁a ▁generating ▁set ▁. ▁In ▁this ▁case ▁we ▁could ▁write ▁, ▁where ▁* ▁denotes ▁the ▁Kle ene ▁star ▁and ▁the ▁mon oid al ▁unit ▁I ▁stands ▁for ▁the ▁empty ▁string . ▁If ▁we ▁start ▁with ▁a ▁set ▁R ▁of ▁generating ▁morph isms ▁( fact s ▁about ▁ ≤ ), ▁we ▁recover ▁the ▁usual ▁notion ▁of ▁semi - Th ue ▁system , ▁where ▁R ▁is ▁called ▁the ▁" rew rit ing ▁rule ". ▁ ▁To ▁return ▁to ▁our ▁example , ▁let ▁N ▁be ▁the ▁category ▁whose ▁objects ▁are ▁the ▁natural ▁numbers ▁ 0 , ▁ 1 , ▁ 2 , ▁... , ▁with ▁a ▁single ▁morph ism ▁ ▁if ▁ ▁in ▁the ▁usual ▁ordering ▁( and ▁no ▁morph isms ▁from ▁i ▁to ▁j ▁otherwise ), ▁and ▁a ▁mon oid al ▁structure ▁with ▁the ▁mon oid al ▁unit ▁given ▁by ▁ 0 ▁and ▁the ▁mon oid al ▁multiplication ▁given ▁by ▁the ▁usual ▁addition , ▁. ▁Then ▁N ▁is ▁a ▁mon oid al ▁pre order ; ▁in ▁fact ▁it ▁is ▁the ▁one ▁freely ▁generated ▁by ▁a ▁single ▁object ▁ 1 , ▁and ▁a ▁single ▁morph ism ▁ 0 ▁ ≤ ▁ 1 , ▁where ▁again ▁