Let $A$ be an artin algebra and let mod$(A)$ be the category of finitely generated $A$-modules.
Let $S$ be a simple module and let $M \in$ mod$(A)$. We further assume that $S$ is a composition factor of $M$. Then there exists a non trivial homomorphism $M \rightarrow I(S)$, where $I(S)$ is the injective envelope of $S$.
What I have so far: We just need to show that there is a non trivial momomorphism $S \rightarrow M$. If we have that, then the existence of a non trivial homomorphism $M \rightarrow I(S)$ follows from the infectivity of $I(S)$.
So let $S$ be a composition factor of $M$. Then there exist submodules $L \subseteq K$ of $M$, such that $S \cong K/L$. Now one needs to construct a non trivial monomorphism $g: K/L \rightarrow M$. Thats the point where I have trouble continuing the proof. Intuitively I would say we construct $g$ by $k+L \mapsto k$. But I don't think it is a well defined.
There is not necessarily a nontrivial monomorphism $S\to M$.
But there is a nontrivial homomorphism $K\to I(S)$: namely, the composition of the obvious maps $K\to K/L\cong S\to I(S)$. And then, by injectivity of $I(S)$, this homomorphism extends to a homomorphism $M\to I(S)$.