Let $F$ is a full subcategory of a category $G$, both categories having binary direct product.
Is it always true that there is such a binary direct product in $G$ that it is a continuation of a binary direct product in $F$?
Hm, can it be generalized for infinitary direct products, rather than only binary? (less important question than the first question above)
Let $F$ be a $4$-element Boolean algebra, say $\{\varnothing,\{0\},\{1\},\{0,1\}\}$ ordered by inclusion. Regard $F$ as a partially ordered set and thus as a category (with a unique morphism $a\to b$ exactly when $a\subseteq b$). Obtain $G$ by inserting into $F$ one new element $z$, above $\varnothing$ but below the other three elements of $F$. Again, regard $G$ as a partially ordered set and thus as a category. Both categories have binary products (i.e., the partially ordered sets have binary meets). But the product of $\{0\}$ and $\{1\}$ in $F$ is $\varnothing$ while in $G$ their product is $z$.