A functor preserves a product of $A$ and $B$ iff $F(A \times B) \cong F(A) \times F(B)$? Let $F \colon \mathbf A \to \mathbf B$ be a functor, and let $A, B \in \mathbf A$. Assume that there exists a product $A \times B$, with projections $p \colon A \times B \to A$ and $q \colon A \times B \to B$. We say that $F$ preserves the product $(A \times B, p, q)$ of $A$ with $B$ if $(F(A \times B), F(p), F(q))$ is a product of $F(A)$ and $F(B)$ in $\mathbf B$.  My question is the following: assuming that a product $F(A) \times F(B)$ exists in $\mathbf B$, is is true that $F$ preserves the product $A \times B$ if and only if $F(A \times B) \cong F(A) \times F(B)$ just as objects in $\mathbf B$? The preservation of the product is equivalent to saying that the canonical morphism $F(A \times B) \to F(A) \times F(B)$ is an isomorphism. So, in other words, I'm asking if from the existence of any isomorphism $F(A \times B) \cong F(A) \times F(B)$ we can deduce that the canonical map is also an isomorphism.
 A: The answer is of course not. 
Consider the trivial case of the constant functor $\Delta(\mathbb N)^{\mathbf {Set}}\colon \mathbf{Set} \to \mathbf {Set}$ defined over $\mathbf {Set}$. This functor sends every set into $\mathbb N$ and every morphism into $1_{\mathbb N}$.
It is well known that $\mathbb N \times \mathbb N \cong \mathbb N$, and so that for every pair of sets $X$ and $Y$ we have that
$$\Delta(\mathbb N)^{\mathbf {Set}}(X \times Y) \cong \Delta(\mathbb N)^{\mathbf {Set}}(X) \times \Delta(\mathbb N)^{\mathbf {Set}}(Y)\ .$$
If your claim where true that we should have that the diagram $(\mathbb N,1_{\mathbb N},1_{\mathbb N})$ is a product in $\mathbf {Set}$, which is not the case.
The point is that in general, if $\mathbf C$ and $\mathbf D$ are categories and $F \colon \mathbf C \to \mathbf D$ is a functor between them, $F$ preserves the product $(A \times B,\pi_A,\pi_B)$ if and only if the morphism $(F(\pi_A),F(\pi_B)) \colon F(A \times B) \to F(A) \times F(B)$, induced by the universal property of the product, is an isomorphism, not simply if there is any isomorphism between the two objects.
