▁convenient ▁to ▁express ▁this ▁fact ▁in ▁arrow ▁form ▁. ▁Because ▁there ▁is ▁at ▁most ▁one ▁such ▁morph ism , ▁we ▁never ▁have ▁to ▁give ▁it ▁a ▁name , ▁such ▁as ▁. ▁The ▁ref lex ivity ▁and ▁trans itivity ▁properties ▁of ▁an ▁order ▁are ▁respectively ▁account ed ▁for ▁by ▁the ▁identity ▁morph ism ▁and ▁the ▁composition ▁formula ▁in ▁C . ▁We ▁write ▁ ▁if f ▁ ▁and ▁, ▁i . e . ▁if ▁they ▁are ▁is omorphic ▁in ▁C . ▁Note ▁that ▁in ▁a ▁partial ▁order , ▁any ▁two ▁is omorphic ▁objects ▁are ▁in ▁fact ▁equal . ▁ ▁Mov ing ▁forward , ▁suppose ▁we ▁want ▁to ▁add ▁a ▁mon oid al ▁structure ▁to ▁the ▁pre order ▁C . ▁To ▁do ▁so ▁means ▁we ▁must ▁choose ▁ ▁an ▁object ▁, ▁called ▁the ▁mon oid al ▁unit , ▁and ▁▁▁ ▁a ▁fun ctor ▁, ▁which ▁we ▁will ▁denote ▁simply ▁by ▁the ▁dot ▁"", ▁called ▁the ▁mon oid al ▁multiplication . ▁Thus ▁for ▁any ▁two ▁objects ▁ ▁we ▁have ▁an ▁object ▁. ▁We ▁must ▁choose ▁ ▁and ▁ ▁to ▁be ▁associ ative ▁and ▁un ital , ▁up ▁to ▁isomorphism . ▁This ▁means ▁we ▁must ▁have : ▁▁▁ ▁and ▁. ▁Furthermore , ▁the ▁fact ▁that ▁· ▁is ▁required ▁to ▁be ▁a ▁fun ctor ▁means — in ▁the ▁present ▁case , ▁where ▁C ▁is ▁a ▁pre order — nothing ▁more ▁than ▁the ▁following : ▁ ▁if ▁ ▁and ▁ ▁then ▁. ▁The ▁additional ▁coh er ence ▁conditions ▁for ▁mon oid al ▁categories ▁are ▁vac uous ▁in ▁this ▁case ▁because