Let $A$ be a finite dimensional algebra and let $M$ be a finite dimensional $A$-module. Then $M$ has a projective cover and I am trying to prove this.
Let $\mathcal{C}$ be a module class and $X$ a module.
A $\mathcal{C}$-precover of $X$ is a morphism $\theta : C \rightarrow X$ with $C \in \mathcal{C}$, such that for any $\theta' : C' \rightarrow X$ with $C' \in \mathcal{C}$, there is $\phi : C' \rightarrow C$ with $\theta' = \theta \phi$.
A $\mathcal{C}$-cover of $X$ is a right minimal $\mathcal{C}$-precover.
I already know that $X$ has a $\mathcal{C}$-cover when it has a $\mathcal{C}$-precover. So I only need to prove that $M$ has a projective precover.
For some $n \in \mathbb{N}$ we can find an epimorphism $\theta: A^n \rightarrow M$. If I now take a morphism $\theta': P \rightarrow M$, $P$ projective, I need to find a morphism $\phi: P \rightarrow A^n$ such that $\theta' = \theta \phi$. How do I get this $\phi$?
And my second questions is, is the proof for the existence of injective envelopes in this case similar?
Thank you in advance!