TITLE: In the category $\mathbf{Set}$ is "the product of an empty set of sets a one-element set"?
QUESTION [7 upvotes]: I was reading these notes on Category Theory and it said (paraphrased to add context):

Exercise 4: Explain why in $\textbf{Set}$ (the Category of Sets), the product of an empty set of sets is a one-element set.

which I think is incorrect. The product of two empty sets (or any number) is empty because we are considering:
$$ \emptyset \times \emptyset = \{ (a,b) : a \in \emptyset, b \in \emptyset \} = \emptyset$$
where $a \in \emptyset , b \in \emptyset$ are false, so the above is the $\emptyset$ which is NOT a one element set (its a zero element set). 
This should be trivial so I am assuming I am somewhere mis reading the natural language of the exercise. Someone help me catch where is it and what the answer should be? i.e. whats being asked and the answer?

REPLY [4 votes]: Since no-one has said this explicitly:
Yes, you are misreading the language of the question. The question is asking for
$$\prod\limits_{x \in \emptyset} x$$
It's the difference between "the product of an empty set of sets" and "the product of a set of empty sets".