While explaining that "In the category of sets the Cartesian product $\prod_{i \in I} A_{i}$ is a product of the family $\{A_{i}: i \in I\}$."
After showing this the author gave the following remark :
But I did not understand why if some $A_{j} = \emptyset,$ then the whole $\prod_{i \in I} A_{i} = \emptyset$ and why there can be no function that satisfies what given in the picture, is not there a mistake in the index j? it must be i instead? could anyone explain this for me please?
$j$ is a particular index for which $A_j$ is the empty set.
Consider an element $x$ in the product. $x$ has a component $x_i$ for each $i\in I$. When it comes to the $j^{th}$ component $x_j$, what can it be? There aren't any options in $A_j$, which is empty, so $x$ can't exist and the product must be empty too.