ctor ▁of ▁taking ▁the ▁dual ▁twice ▁repeated , ▁is ▁is omorphic ▁to ▁the ▁identity ▁fun ctor , ▁i . e . ▁a ▁p iv otal ▁category . ▁Then ▁for ▁any ▁right ▁rig id ▁object ▁X , ▁and ▁any ▁other ▁object ▁Y , ▁we ▁may ▁define ▁the ▁isomorphism ▁ ▁and ▁its ▁reci pro cal ▁isomorphism ▁ ▁. ▁ ▁Then ▁for ▁any ▁end omorphism ▁, ▁the ▁trace ▁is ▁of ▁f ▁is ▁defined ▁as ▁the ▁composition : ▁ ▁We ▁may ▁continue ▁further ▁and ▁define ▁the ▁dimension ▁of ▁a ▁rig id ▁object ▁to ▁be : ▁. ▁ ▁R ig id ity ▁is ▁also ▁important ▁because ▁of ▁its ▁relation ▁to ▁internal ▁Hom ' s . ▁If ▁X ▁is ▁a ▁left ▁rig id ▁object , ▁then ▁every ▁internal ▁Hom ▁of ▁the ▁form ▁[ X , ▁Z ] ▁exists ▁and ▁is ▁is omorphic ▁to ▁Z ▁ ⊗ ▁Y . ▁In ▁particular , ▁in ▁a ▁rig id ▁category , ▁all ▁internal ▁Hom ' s ▁exist . ▁ ▁Altern ative ▁termin ology ▁ ▁A ▁mon oid al ▁category ▁where ▁every ▁object ▁has ▁a ▁left ▁( resp . ▁right ) ▁dual ▁is ▁also ▁sometimes ▁called ▁a ▁left ▁( resp . ▁right ) ▁autonom ous ▁category . ▁ ▁A ▁mon oid al ▁category ▁where ▁every ▁object ▁has ▁both ▁a ▁left ▁and ▁a ▁right ▁dual ▁is ▁sometimes ▁called ▁an ▁autonom ous ▁category . ▁ ▁An ▁autonom ous ▁category ▁that ▁is ▁also ▁symmetric ▁is ▁called ▁a ▁compact ▁closed ▁category . ▁ ▁Disc uss ion ▁ ▁A ▁mon oid al ▁category ▁is ▁a ▁category ▁with ▁a ▁tensor