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?



Is the Cartesian product of two finitely presented objects finitely presentable?
No. For counterexamples, see Theorems 3.8, 3.9, and 3.10 of 
Finiteness properties of direct products
of algebraic structures  

Peter Mayr, Nik Ruškuc  

Journal of Algebra 494 (2018) 167-187.
These theorems show that products of finitely presented loops, idempotent magmas, or lattices may fail to be finitely presentable.






