reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem Th6:
  for C being strict Category, A being strict Subcategory of C st
  the carrier of A = the carrier of C & the carrier' of A = the carrier' of C
  holds A = C
proof
  let C be strict Category, A be strict Subcategory of C such that
A1: the carrier of A = the carrier of C and
A2: the carrier' of A = the carrier' of C;
A3: the Target of A = (the Target of C)|the carrier' of A by Th4
    .= the Target of C by A2;
A4: the Comp of A = (the Comp of C)||the carrier' of A by Th4
    .= the Comp of C by A2;
  the Source of A = (the Source of C)|the carrier' of A by Th4
    .= the Source of C by A2;
  hence thesis by A1,A3,A4;
end;
