In some old notes I found the following formula for the cardinality of the cardinality of objects and arrows in a product category:
$$ |Ob(C \times D)| = |Ob(C)| \cdot |Ob(D)| \\ |Hom(C \times D)| = |Ob(C)| \cdot |Hom(D)| + |Ob(D)| \cdot |Hom(C)| $$
I also noted that the formula for arrows excludes identities.
While the formula for the formula for objects seems obviously true to me, I can't really confirm the second formula. E.g. when I write down the product category of the free quiver, I only get 12 arrows.
Is the formula wrong? I tried searching for it, but I didn't find it again.