Is the following true?
Lemma. Let $F:C \to C$ be a functor that preserves products, i.e. for all $A, B \in C$ we have isomorphism $\alpha_{A,B} : F(A\times B) \cong FA \times FB$ and this isomorphism commutes with product projections - this means $Fp_A \alpha_{A,B}=p_{FA}$.
Fix $B \in C$.
Then $\alpha_{A,B}$'s for each $A$ form a natural isomorphism: $\alpha_{B}:F(-)\times FB \Rightarrow F(- \times B)$.
More generally, whenever you have bifunctors $F,G : A\times B\to C$ and a natural transformation $\eta : F\to G$ it is easy to see that when you fix $b\in B$, $(\eta_{a,b})_{a\in A}$ is a natural transformation $F(-,b)\to G(-,b)$
Of course if $\eta$ is an isomorphism, this will be an isomorphism too