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 Th39:
  a is initial & dom f = a & cod f = b implies init(a,b) = f
proof
  assume that
A1: a is initial and
A2: dom f = a & cod f = b;
  consider h being Morphism of a,b such that
A3: for g being Morphism of a,b holds h = g by A1;
  f is Morphism of a,b by A2,CAT_1:4;
  hence f = h by A3
    .= init(a,b) by A3;
end;
