Suppose $C,D$ are (additive) categories and $F:C\rightarrow D$ is a functor. I want to know when is $F(\alpha\cdot C)=F(\alpha)\cdot F(C)$ where $\alpha\cdot C$ denotes the image of $C$ under the morphism $\alpha$. Is it true in general or do we have to impose restrictions on the categories or the functor? I tried to prove it as it seems to be a valid assertion but couldn't get anywhere. In my particular case, $C=D$ and $\alpha\in\operatorname{End}(C)$ is an idempotent morphism.
I am using the following definitions of images and kernels in additive categories. Suppose $A,B$ are objects in an additive category $C$ and $f\in\operatorname{Hom}(A,B)$.
The $Ker(f)$ is an object with a morphism $i:Ker(f)\rightarrow A$ such that the following sequence is exact: $$ 0\rightarrow \operatorname{Hom}(-,Ker(f))\xrightarrow{i\circ-}\operatorname{Hom}(-,A)\xrightarrow{f\circ-}\operatorname{Hom}(-,B) $$ The $coker(f)$ is an object with a morphism $q:B\rightarrow coker(f)$ such that the following sequence is exact: $$ 0\rightarrow \operatorname{Hom}(coker(f),-)\xrightarrow{-\circ q}\operatorname{Hom}(B,-)\xrightarrow{-\circ f}\operatorname{Hom}(A,-) $$ $Im(f)$ is defined to be $Ker(q)$ and $coim(f)$ is defined to be $coker(i)$.
Does anyone know how to prove that $F(\alpha\cdot C)=F(\alpha)\cdot F(C)$ under the relevant restrictions?