$k$-linear category

Question

Let $C$ be a additive category and $k$ is a commutative ring. $C$ is called $k$-linear if the morphism sets $C(x,y)$ have $k$-module structures for all $x,y\in Obj(C)$ and the compositions of morphisms are $k$-bilinear maps. It is said in the paper "Rings with several objects" by Mitchell that " $C$ is $k$-linear if and only if there exists a ring-homomorphism from $k$ to the center $Z(C)$ of $C$".

Every element $η∈Z(C)$ is given by a family of endomorphisms $η_x:x→x$ where $x∈C$, such that for all morphisms $x→y$ the obvious diagram commutes. So for every morphism $f:x→y$ in $C(x,y)$ we can define the multiplication $\eta *f:=\eta_y\circ f$. So I can only check the "if" part of the above statement, please help me to check the "only if" part.

I feel that I'm lack of category background for reading the paper "Rings with several objects" by Mitchell, would you give me some advice about which category books or papers I should take a look.

Thank you very much!

Answer 1

If $C$ carries a $k$-linear structure, then define $k \to Z(C)$ by $\lambda \to (x \mapsto \lambda \mathrm{id}_x)$. One checks that this is a homomorphism of rings. If conversely, $\phi : k \to Z(C)$ is a homomorphism of rings, then define a $k$-linear structure by $\lambda f := f \phi(\lambda)(x) = \phi(\lambda)(y) f$ for $f : x \to y$ and $\lambda \in k$. One checks that this is in fact a $k$-linear structure. One also checks that these constructions are inverse to each other. In fact, this is some kind of Yoneda lemma ("by naturality it suffices to look at the identity"). The verifications are straight forward. Apart from the definitions, no specific knowledge on categories has to be used.