direct products of homomorphisms

Assume that ${\{f_i:G_i\to H_i\}}_{i\in I}$ is a family of homomorphisms between groups. Then we can define the Cartesian product (or unrestricted direct product) of this family as a homomorphism

$$\prod _{i\in I}{f}_i:\prod _{i\in I}{G}_i\to \prod _{i\in I}{H}_i$$

such that

$$\left(\prod _{i\in I}{f}_i\right)\left(g\right)(j)=f_j(g(j))$$

for each $g\in {\prod }_{i\in I}{G}_i$ and $j\in I$.

One can easily show that ${\prod }_{i\in I}{f}_i$ is a group homomorphism. Moreover it is clear that

$$\left(\prod _{i\in I}{f}_i\right)\left(\underset{i\in I}{\oplus }{G}_i\right)\subseteq \underset{i\in I}{\oplus }{H}_i,$$

so ${\prod }_{i\in I}{f}_i$ induces a homomorphism

$$\underset{i\in I}{\oplus }{f}_i:\underset{i\in I}{\oplus }{G}_i\to \underset{i\in I}{\oplus }{H}_i,$$

which is a restriction of ${\prod }_{i\in I}{f}_i$ to ${\oplus }_{i\in I}{G}_i$. This homomorphism is called the direct product (or restricted direct product) of ${\{f_i:G_i\to H_i\}}_{i\in I}$.