I'm trying to generalize one aspect of the accepted answer in: $\text{Hom}(\mathbb{F}_p G, M)$ and $H^1(G,M)$
Is the following statement correct?
If:
- $R_1$ and $R_2$ are rings;
- $F:R_1-\text{mod}\rightarrow R_2-\text{Mod}$ is an additive functor;
- $M \in R_1-\text{Mod}$.
Then:
- $F(M)$ is a $\text{Hom}_{R_1}(M,M)$-module where
- the action of $\text{Hom}_{R_1}(M,M)$ on $F(M)$ is given by $f\cdot x=F(f)(x)$.