Is free product of groups always bigger that direct product? My question is as follow:

Let $\{G_\alpha\}, \alpha \in A$ be a class of groups. Is it always true that there exists a surjective homomorphism $\phi$
$$\phi:*_\alpha \,G_\alpha\, \to \prod_\alpha G_\alpha.\,$$

For example, there is a surjective homomorphism from $\mathbb{Z}*\mathbb{Z}$ to $\mathbb{Z} \times \mathbb{Z}$.

And how to type the * bigger?
Can you have a surjective homomorphism from
$$
 \underset{{i\in\mathbb{N}}}{{\LARGE{\ast}}}\mathbb{Z}_2 \to \prod_{i\in\mathbb{N}}\mathbb{Z}_2 ?
$$What would be the image of $(1,1,1,1,\ldots)   $?
Can you have a surjective homomorphism from
$$
 \underset{{i\in\mathbb{N}}}{{\LARGE{\ast}}}\mathbb{Z}_2 \to \prod_{i\in\mathbb{N}}\mathbb{Z}_2 ?
$$What would be the image of $(1,1,1,1,\ldots)   $?