Example of a functor on products

Question

I am trying to come up with an example of a functor which maps products to products but not the same one. That is:

Let $C,D$ be categories such that $C$ has all products. I need to define $T:C\rightarrow D$ such that for two objects in $C$ we have $U\times V$ and $T(U\times V)$ and $T(U)\times T(V)$ are both products but not the same product.

Thanks for the help

Answer 1

Let $C$ and $D$ be the category of sets. Choose a nonempty set, say $A$, and define the functor $T$ by $T(U)=U\times A$. When $f:U\to V$ is a morphism, $T(f)=f\times 1_A$. You can check that $T$ is a functor.

Now we have $T(U\times V) = (U\times V)\times A$ but $T(U)\times T(V) = (U\times A)\times(V\times A)$; both are products, but in general are not equal.