I would love to know if my answer to the following question is correct, this feels weird to me, because the statement of the question sounds very correct and intuitive, but if you don't consider a special case you get it wrong. This is the question:
$$A {\times}B \sqsubseteq C {\times}D \leftrightarrow (A\sqsubseteq C, B\sqsubseteq D)$$
The special case is when one of the groups is $\emptyset$, For example:
$$A=\emptyset, B=\mathbb{N}, C=\{1\}, D=\{2\}$$.
In that case, The statement on the left holds but the statement on the right is false, because $B\sqsubseteq D$ does not hold.
Could someone please tell me if I'm right or wrong?
You are right. The given implication holds under the assumption that sets there are non-empty but not when empty set is involved.