Let $\Lambda$ be an artin algebra. We deonte by $mod \Lambda$ the category of all finitely generated $\Lambda$-modules. $\mathcal{Y}$ is a subcategory of $mod \Lambda$ closed under extensions and summands.
I have seen the following :"If $Ext_{\Lambda} ^1(C,-) |_{\mathcal{Y}}$ is a finitely generated functor from $\mathcal{Y}$ to abelian groups, since $\mathcal{Y}$ is closed under summands it follows that $Ext_{\Lambda} ^1(C,-) |_{\mathcal{Y}}$ has a projective cover $Hom_{\Lambda}(Y,-) \stackrel{\varphi}{\longrightarrow} Ext_{\Lambda} ^1(C,-) |_{\mathcal{Y}}$. Let $0 \rightarrow Y \rightarrow X \stackrel{f}{\longrightarrow} C \rightarrow 0$ be $\varphi(1_X)$. Then $f:X \rightarrow C$ is right minimal is an immediate consequence of the fact that $\varphi: Hom_{\Lambda}(Y,-) \rightarrow Ext_{\Lambda} ^1(C,-) |_{\mathcal{Y}}$ is a projective cover"
I just know finitely generated or projective cover for modules, could anyone provide me some references about those theories about functors? Or how to get that $f: X \rightarrow C$ is right minimal?