Let $\Lambda$ be an artin algebra of finite type. $M$ is the direct sum of all nonisomorphic indecomposable $\Lambda$-modules, $\Gamma_M :=End_{\Lambda}(M)$. Then we know there is an equivalence $Hom_{\Lambda}(M,-): mod \Lambda \rightarrow P(\Gamma_M)$(Here $mod\Lambda$ is the category of finitely generated $\Lambda$-modules, $P(\Gamma_M)$ is the category of projective $\Gamma_M$-modules). We also know that for modules over rings, surjective homomorphism and epimorphism are same.
Now let $S$ be a non-projective $\Lambda$-module, $P(S)$ is the projective cover of $S$. Then there is a surjective morphism: $P \rightarrow S \rightarrow 0$, by the functor $Hom_{\Lambda}(M,-)$, we have $Hom_{\Lambda}(M,P) \rightarrow Hom_{\Lambda}(M,S) \rightarrow 0$ in $P(\Gamma_M)$ since $Hom_{\Lambda}(M,-)$ is an equivalence. But we know $Hom_{\Lambda}(M,P) \rightarrow Hom_{\Lambda}(M,S)$ can not be surjective in $mod \Gamma_M$ (otherwise we can get $S$ is a direct summand of $P(S)$ since $Hom_{\Lambda}(M,S)$ is a projective $\Gamma_M$-module). So I get a morphism which is surjective in subcategory $P(\Gamma_M)$ but not surjective in $mod\Gamma_M$.
But I think that for a morphism, if it is surjective in subcategory $P(\Gamma_M)$, we can also get it is surjective in the module category $mod \Gamma_M$. So who can tell me where is wrong?