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 Th68:
  for F be Function of I,the carrier' of C holds F is
  Projections_family of c,I iff F opp is Injections_family of c opp,I
proof
  let F be Function of I,the carrier' of C;
  thus F is Projections_family of c,I implies F opp is Injections_family of c
  opp,I
  proof
    assume
A1: doms 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;
    then
A3:  gg opp = (F/.x)opp by OPPCAT_1:def 6;
      assume
A4:   x in I;
      hence (cods(F opp))/.x = cod((F opp)/.x) by Def2
        .= cod((F/.x) opp) by A4,Def3
        .= dom(F/.x) opp by A2,A3,OPPCAT_1:12
        .= (I --> (c opp))/.x by A1,A4,Def1;
    end;
    hence cods(F opp) = I --> (c opp) by Th1;
  end;
  assume
A5: cods(F opp) = I --> (c opp);
  now
   let x;
    reconsider gg = F/.x as Morphism of dom(F/.x), cod(F/.x) by CAT_1:4;
A6:  Hom(dom gg,cod gg) <> {} by CAT_1:2;
    then
A7:  gg opp = (F/.x)opp by OPPCAT_1:def 6;
   assume
A8: x in I;
    hence (doms F)/.x = dom(F/.x) by Def1
      .= cod(gg opp) by A6,OPPCAT_1:10
      .= cod((F opp)/.x) by A8,Def3,A7
      .= (I --> c)/.x by A8,A5,Def2;
  end;
  hence doms F = I --> c by Th1;
end;
