Are these particular cases of the universal property of the coproduct and product?

Question

I was talking to a friend about the results presented here and here.
He told me that these are particular cases of the universal property of the coproduct and of the product (in a category). I have learnt a bit about universal properties, but I can't see why what he says is true. For instance, those universal properties (that of a direct product of groups, for instance) don't mention anything about having isomoprhisms in some cases. So, could someone enlighthen me a bit?

Answer 1

The point is that the Cartesian product $A\times B$ of two groups $A,B$ with the projection maps $\pi_A,\,\pi_B$ satisfies the universal property:
whenever we have a similar diagram, i e. homomorphisms $f:C\to A,\ g:C\to B$, then there's a unique homomorphism $u:C\to A\times B$ which makes $$f=\pi_A\circ u\ \text{ and }\ g=\pi_B\circ u\,.$$ Namely, this condition forces $u(c)=(f(c),\,g(c))$, and this also works for defining $u$.

One more important thing is that whenever an object $C$ with maps $f:C\to A,\ g:C\to B$ also satisfies the universal property, then $u$ above turns out to be an isomorphism, hence $C\cong A\times B$.