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 $$\bigg( \prod_{i\in I}f_i \bigg) \big( g\big) (j)=f_j(g(j))$$ 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 $$\bigg( \prod_{i\in I}f_i \bigg) \big( \bigoplus_{i\in I} G_i \big) \subseteq \bigoplus_{i\in I} H_i,$$ so $\prod_{i\in I}f_i$ induces a homomorphism $$\bigoplus_{i\in I}f_i:\bigoplus_{i\in I}G_i\to \bigoplus_{i\in I}H_i,$$ which is a restriction of $\prod_{i\in I}f_i$ to $\bigoplus_{i\in I}G_i$ This homomorphism is called the direct product (or restricted direct product) of $\{f_i:G_i\to H_i\}_{i\in I}$