reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;
reserve f for Morphism of a,b,
        g for Morphism of b,a;
reserve g for Morphism of b,c;
reserve f,g for Morphism of C;

theorem Th69:
  for F be Function of I,the carrier' of C opp, c being Object of
  C opp holds F is Injections_family of c,I iff opp F is Projections_family of
  opp c,I
proof
  let F be Function of I,the carrier' of C opp, c be Object of C opp;
  thus F is Injections_family of c,I implies opp F is Projections_family of
  opp c,I
  proof
    assume
A1: cods F = I --> c;
    now
      let x;
    reconsider gg = F/.x as Morphism of dom(F/.x), cod(F/.x) by CAT_1:4;
A2:  Hom(dom gg,cod gg) <> {} by CAT_1:2;
      assume
A3:   x in I;
      hence (doms opp F)/.x = dom(opp F/.x) by Def1
        .= dom(opp(F/.x)) by A3,Def4
        .= opp(cod(F/.x)) by A2,OPPCAT_1:13
        .= (I --> (opp c))/.x by A1,A3,Def2;
    end;
    hence doms opp F = I --> (opp c) by Th1;
  end;
  assume
A4: doms opp F = I --> (opp c);
  now
    let x;
    reconsider gg = F/.x as Morphism of dom(F/.x), cod(F/.x) by CAT_1:4;
  Hom(dom gg,cod gg) <> {} by CAT_1:2;
    then
A5:  gg opp = (F/.x)opp by OPPCAT_1:def 6;
    assume
A6: x in I;
    hence (cods F)/.x = cod(F/.x) by Def2
      .= dom(opp(F/.x)) by A5,OPPCAT_1:11
      .= dom(opp F/.x) by A6,Def4
      .= (I --> c)/.x by A4,A6,Def1;
  end;
  hence cods F = I --> c by Th1;
end;
