I am working on a paper in which I need to talk about what I call concrete categories with standard products. I write $U(X)$ for the underlying set of an object $X$ in a concrete category $\cal C$ and say $\cal C$ has standard products, if for any two objects $X$ and $Y$, there is an object $X \times Y$ of $\cal C$, such that (i) $U(X \times Y)$ is the set-theoretic product $U(X)\times U(Y)$, (ii) the set-theoretic projections of $U(X \times Y) = U(X) \times U(Y)$ onto its factors are $\cal C$-morphisms and (iii) if $f : Z \to X$ and $g : Z \to Y$ are $\cal C$-morphisms, then $\langle f, g\rangle : U(Z) \to U(X \times Y)$ is a $\cal C$-morphhism.
Many concrete categories have standard products, e.g., topological spaces, groups (or any other variety of algebras). Many concrete categories don't have products at all, e.g., fields, integral domains. I would like a natural example of a concrete category that has products but not standard products. The best I have come up with is the opposite of a concrete category like groups whose coproducts are not products. But that means I have to explain to my readers the device that makes the opposite of a concrete category into a concrete category. I would be very grateful for a nicer example, if anyone has one.