I want to show that if projective covers exist then they are unique up to isomorphism.
More precisely let $f: P \rightarrow M$ and $g: Q \rightarrow M$ be projective covers of an $R$-module $M$.
Using the fact that $g$ is surjective and that $P$ is projective we can find an $R$-map $h: P \rightarrow Q$ such that $g \circ h=f$.
Note then that $\operatorname{Im}(h)+\operatorname{ker}(g)=Q$. Since $ker(g)$ is superfluous this implies that $Q=Im(h)$ so that $h$ is surjective.
But how do we conclude that $h$ is injective?