TITLE: Can we turn the functor "category ring" into a 2-functor in a natural way?
QUESTION [9 upvotes]: Let $C$ be a small pre-additive category. Let $R(C)$ denote its category ring, that is,
$$
R(C)=\bigoplus_{a,b\in \mathrm{Ob}(C)} C(a,b)
$$
as Abelian group, where the direct sum runs over all object $a$, $b$ of $C$. The multiplication in $R(C)$ is given by composition of composable morphisms and 0 for uncomposable morphisms (extended by bilinearity).
This constuction is functorial: An additive functor $C\to D$ between small pre-additive categories induces a ring homomorphism $R(C)\to R(D)$ between the corresponding category rings in a canocical way.
Hence we have a functor $R$ from the category of small pre-additive categories (with additive functors as morhisms) to the category of rings (with ring homomorphisms as morphisms).
But the category of small pre-additive categories has a 2-categorical structure given by natural transformations.
Hence my question: Does this 2-categorical structure have a counterpart in the category of rings? More precisely, is there a natural notion of 2-morphisms between ring homomorphisms turning $R$ into a 2-functor?

REPLY [3 votes]: This construction is not even a functor!
The problem is that two morphisms which are not composable in $C$ may become composable in $D$. The simplest example is to take $C$ to be the discrete category on two objects $a, b$, $D$ to be the discrete category on an object $d$, and $F : C \to D$ the unique functor between them (or rather the free preadditive category on these). Then $R(C)$ has two generators $\text{id}_a$ and $\text{id}_b$ which are not composable and hence whose product is $0$. But their image in $D$ is $\text{id}_d$ which squares to itself. So
$$R(F)(\text{id}_a \times \text{id}_b) = R(F)(0) = 0 \neq R(F)(\text{id}_a) \times R(F)(\text{id}_b) = \text{id}_d.$$
So $R(F)$ is not a ring homomorphism.
This construction is not a functor, but it can be understood in terms of a functor as follows. I'll restrict my attention to the case that $C$ has finitely many objects. Given such a preadditive category you can talk about the category obtained by formally adjoining finite direct sums (this is a functor, even a 2-functor), and in this new category I claim that the category ring is just $\text{End}(\oplus_{c \in C} c)$. The significance of this ring is that it is Morita equivalent to $C$, meaning that right modules over it are naturally equivalent to presheaves on $C$ valued in abelian groups.