What remains to be shown now is that gcd(m,n) satisfies the universal property. First we'll restate the universal property for a general category, and then instantiate it to the integer category to prove that gcd(m,n) satisfies it.
But here's a motivating hint for the universal property, and why it is needed. In the above construction that I gave, I proved the existence of the projection morphisms \\(\pi_1\\) and \\(\pi_2\\). But note that that argument only depended upon the fact that gcd(m,n) was a common divisor of m and n. I didn't invoke the fact that gcd(m,n) is the _greatest_ common divisor of m and n. That _greatest_ property is exactly what is needed to prove that gcd(m,n) satisfies the universal property.