In $\mathbf{Grp}$, the finitely presentable objects are precisely the finitely presented groups. Let $F_2$ be the free group on two elements. Then [$F_2 \times F_2$ is finitely generated but not finitely presented](https://math.stackexchange.com/questions/35579/non-finitely-presented-subgroup), so the class of finitely presentable objects in $\mathbf{Grp}$ is not closed under binary products.