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 the $k$-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!