Let ${\cal A}$ and ${\cal B}$ be abelian categories. Let $(F^n, \delta_F)$, $(G^n, \delta_G)$ be $\delta$-functors from ${\cal A}$ to ${\cal B}$. A morphism of $\delta$-functors from $F$ to $G$ is a collection of transformation of functors $t_n \colon F^n \to G^n$, $n > 0$ such that for every short exact sequence $0 \to A \to B \to C \to 0$ of ${\cal A}$ the diagrams are commutative.
The universal $\delta$-functor $(F^n, \delta_F)$ satisfies that for any morphism of functors $t \colon F^0 \to G^0$ there exists a unique morphism of $\delta$-functors $\{t^n\}_{n \geq 0} \colon F^n \to G^n$ such that $t_0 = t$.
Suppose that $M$ is a $G$-module such that $pM = 0$, i.e. $p$-torsion. Then I would like to show that for any $n \geq 0$, $p H^n(G, M) = 0$, i.e., $H^n(G, M)$ is a $p$-torsion module.
Q. How can I apply the universality of the $\delta$-functor $(H^n, \delta_H)$, to prove that $p H^n(G, M) = 0$ for $n \geq 0$?