reserve x,y for set;

theorem Th1:
  for A,B being transitive with_units non empty AltCatStr for F
being feasible reflexive FunctorStr over A,B st F is coreflexive bijective for
  a being Object of A, b being Object of B holds F.a = b iff F".b = a
proof
  let A,B be transitive with_units non empty AltCatStr;
  let F be feasible reflexive FunctorStr over A,B such that
A1: F is coreflexive bijective;
  reconsider G = F" as feasible reflexive FunctorStr over B,A by A1,FUNCTOR0:35
,36;
  let a be Object of A, b be Object of B;
  F" * F = id A by A1,FUNCTOR1:19;
  then a = (F" * F).a by FUNCTOR0:29;
  hence F.a = b implies F".b = a by FUNCTOR0:33;
  F * G = id B by A1,FUNCTOR1:18;
  then b = (F * G).b by FUNCTOR0:29;
  hence thesis by FUNCTOR0:33;
end;
