▁is ▁easily ▁shown ▁to ▁satisfy ▁the ▁requ is ite ▁universal ▁property . ▁ ▁See ▁also ▁almost ▁periodic ▁function . ▁ ▁C ategor ical ▁consider ations ▁▁ ▁Pon try agin ▁du ality ▁can ▁also ▁profit ably ▁be ▁considered ▁fun ctor ial ly . ▁ ▁In ▁what ▁follows , ▁L CA ▁is ▁the ▁category ▁of ▁locally ▁compact ▁ab elian ▁groups ▁and ▁continuous ▁group ▁hom omorph isms . ▁The ▁dual ▁group ▁construction ▁of ▁ ▁is ▁a ▁cont rav ari ant ▁fun ctor ▁L CA ▁→ ▁L CA , ▁represented ▁( in ▁the ▁sense ▁of ▁represent able ▁fun ctors ) ▁by ▁the ▁circle ▁group ▁ ▁as ▁▁ ▁In ▁particular , ▁the ▁double ▁dual ▁fun ctor ▁ ▁is ▁cov ari ant . ▁A ▁categor ical ▁form ulation ▁of ▁Pon try agin ▁du ality ▁then ▁states ▁that ▁the ▁natural ▁transformation ▁between ▁the ▁identity ▁fun ctor ▁on ▁L CA ▁and ▁the ▁double ▁dual ▁fun ctor ▁is ▁an ▁isomorphism . ▁Un w inding ▁the ▁notion ▁of ▁a ▁natural ▁transformation , ▁this ▁means ▁that ▁the ▁maps ▁ ▁are ▁is omorph isms ▁for ▁any ▁locally ▁compact ▁ab elian ▁group ▁G , ▁and ▁these ▁is omorph isms ▁are ▁fun ctor ial ▁in ▁G . ▁This ▁isomorphism ▁is ▁analog ous ▁to ▁the ▁double ▁dual ▁of ▁finite - dimensional ▁vector ▁spaces ▁( a ▁special ▁case , ▁for ▁real ▁and ▁complex ▁vector ▁spaces ). ▁ ▁An ▁immediate ▁consequence ▁of ▁this ▁form ulation ▁is ▁another ▁common ▁categor ical ▁form ulation ▁of ▁Pon try agin ▁du ality : ▁the ▁dual ▁group ▁fun ctor ▁is ▁an ▁equivalence ▁of ▁categories ▁from