This seems very obvious, and that may be the reason why I cannot find this online. I think it's yes, and my proof is that the isomorphism between the products is the product of the isomorphisms: Given
$$\varphi_1: A \to B,\ \varphi_2: C \to D,$$
define $$\varphi(a \times c) = \varphi_1(a) \times \varphi_2(c)$$
and then show that $\varphi$ is injective, surjective and a homomorphism, all of which follow from injectivity, surjectivity and homomorphism property of $\varphi_1$ and $\varphi_2$.
First question, am I correct?
Second question, what's the relation between the above, this exercise from Artin Algebra, this exercise from Munkres Topology and this theorem from Munkres Topology?
- Exercise 2.11.8 in Artin Algebra
- Exercise 18.10 in Munkres Topology
- Theorem 18.4 ("Maps into products") in Munkres Topology:
By the way, the context for my questions is this theorem in Munkres Topology:
- Theorem 67.8 in Munkres Topology