Isomorphism of Direct Product of Groups

Question

I have $H_1,H_2,\dots, H_n$ groups with the property $H_i\cong G_i$, where $G_1,\dots,G_n$ are also groups.

It should be somehow easily followed that $G_1\times \dots\times G_n\cong H_1\times \dots\times H_n$.

I would define a function $\phi:G_1\times\dots\times G_n\to H_1\times \dots\times H_n$ where $\phi(g)_i=h_i$ and prove that it is an bijective homomorphism which should be clear for that function, but is this enough?

Answer 1

Denote by $\varphi_i$ the isomorphism betwenn $G_i$ and $H_i$. The function $\prod_{i=1}^n\varphi_i$ is the function which sends $(g_1,...,g_n)\in G_1\times...\times G_n$ to $(\varphi_1(g_1),...,\varphi_n(g_n))\in H_1\times ...\times H_n$. Check that this function is injective and surjective.