Let's say we have a locally $\lambda$-presentable category and a pair of $\lambda$-presentable objects $A$ and $B$. Is it true that $A \times B$ is $\lambda$-presentable?



Cartesian product of small objects
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.


