I'm new to projective resolutions, derived functors, etc, so I'd like a proof check of my proof that the derived functors of an exact contravariant functor are trivial.
Suppose that $F$ is an exact contravariant functor and let $M$ be a module over some commutative ring $R$. Take a projective resolution of $M$, $$ \cdots \to P_2 \to P_1 \to P_0 \twoheadrightarrow M. $$ Remove the term $M$, and apply $F$ to obtain the chain complex \begin{equation}\label{eq} F(P_0) \to F(P_1) \to F(P_2) \to \cdots. \end{equation} This sequence is exact, since $F$ is. It therefore has trivial homology, which shows that all right derived functors of $F$ are $0$.
To show that all left derived functors of $F$ are $0$, we instead take an injective resolution of $M$ and follow the same procedure. EDIT: I included this, just for the sake of completeness:
Let $$ M \hookrightarrow I_0 \to I_1 \to I_2 \to \cdots $$ be an injective resolution of $M$. Again, remove the term $M$ and apply $F$ to obtain $$ \dots \to F(I_2) \to F(I_1) \to F(I_0). $$ Just as before, this complex has trivial homology.
Any corrections or tips are greatly appreciated!