let I be set ; :: thesis: for C being Category
for F being Function of I, the carrier' of C holds (F opp) opp = F
let C be Category; :: thesis: for F being Function of I, the carrier' of C holds (F opp) opp = F
let F be Function of I, the carrier' of C; :: thesis: (F opp) opp = F

now :: thesis: ( the carrier' of C = the carrier' of ((C opp) opp) & ( for x being set st x in I holds
((F opp) opp) /. x = F /. x ) )

thus the carrier' of C = the carrier' of ((C opp) opp) ; :: thesis: for x being set st x in I holds
((F opp) opp) /. x = F /. x
let x be set ; :: thesis: ( x in I implies ((F opp) opp) /. x = F /. x )
assume A1: x in I ; :: thesis: ((F opp) opp) /. x = F /. x
hence ((F opp) opp) /. x = ((F opp) /. x) opp by Def3
.= ((F /. x) opp) opp by A1, Def3
.= F /. x ;
:: thesis: verum

end;

hence (F opp) opp = F by Th1; :: thesis: verum