Let $C$ be a locally finitely presented category, $A, B$ are two finitely presented (synonym: compact) objects in it. Is it true that $A \times B$ is finitely representable?

At least I have looked at a number of examples of categories of algebras of (finitary) algebraic theories and this seems to be true (although, say, for the category of semigroups the rule for constructing the desired presentation looks unclear). On the other hand, I'm not sure whether this is true for any topos of presheaves?

internally finitely presentableobjects as those objects $A$ for which the internal hom functor $-^A$ preserves filtered colimits. Then, $-^{A\times B}\cong(-^A)^B$ so product of two internally finitely presentable objects is internally finitely presentable. You can next externalize this using $\hom(A,-)=\Gamma(-^A)$providedthe global elements functor $\Gamma=\hom(1,-)$ preserves filtered colimits. This is related to compactness of the topos itself, that is, it will work if the terminal object $1$ is (externally) finitely presentable. $\endgroup$