Functors on the category of abelian groups which satisfy $F(G\times H) \cong F(G)\otimes_{\mathbb{Z}} F(H)$ Edit: According to the comment of Todd Trimble, I revise the question.
What are some examples of functors $F$ on the category of Abelian groups or category of rings which satisfy $$F(G\times H)\cong F(G)\otimes_{\mathbb{Z}} F(H)$$
 A: *

*The first thing that comes to mind is the symmetric algebra functor
$$A \mapsto Sym_{\mathbb Z}(A) = \oplus_{n \in \mathbb N} Sym^n_{\mathbb Z}(A) = \mathbb Z \oplus A \oplus (A\otimes A/ \Sigma_2) \oplus \dots $$
This may be regarded as a functor from abelian groups to commutative rings (it is left adjoint to the forgetful functor); or by composing with the forgetful functor it may be regarded as an endofunctor of abelian groups.
One way to think about this is to observe that $\oplus$ is the binary coproduct of abelian groups and $\otimes$ is the binary coproduct of commutative rings, so any left adjoint functor $Ab \to CRing$ must take $\oplus$ to $\otimes$.
Of course, many variants immediately present themselves. One may precompose with any additive functor and postcompose with any monoidal functor. For instance, $A \mapsto Sym_{\mathbb Q}(A) = \mathbb Q \oplus (\mathbb Q \otimes A) \oplus \dots$ also has the desired property.


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*For another example, take the group ring functor $A \mapsto \mathbb Z[A]$, which again can be regarded as either a functor $Ab \to CRing$ or $Ab \to Ab$. Here $\mathbb Z[A]$ is the free abelian group on the underlying set of $A$; if $e_a,e_b$ are the generators corresponding to $a,b \in A$, then we define $e_a e_b = e_{a+b}$. This has the desired for the same reason as $Sym_{\mathbb Z}$: regarded as a functor $Ab \to CRing$, it is left adjoint to the "group of units" functor $R \mapsto R^\times$.


Again there are variants like $A \mapsto \mathbb Q[A]$.


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*More generally, if $G$ is any affine abelian group scheme over $Spec \mathbb Z$, then $Hom(\Gamma(G),-): CRing \to Ab$ is a right adjoint functor, whose left adjoint has the desired property.

