Let $\Lambda$ be an artin algebra and $mod \Lambda$ be the category of finitely generated $\Lambda$-modules. Suppose $\mathcal{Y}$ is a subcategory of $mod\Lambda$ which is closed under extension and direct summand and $C \in mod \Lambda$.
If $Ext_{\Lambda}^1(C,-)|_{\mathcal{Y}}$ which means restricted to $\mathcal{Y}$ is finitely generated and $f:(Y,-)|_{\mathcal{Y}} \rightarrow Ext_{\Lambda}^1(C,-)|_{\mathcal{Y}}$ is the projective cover of $Ext_{\Lambda}^1(C,-)|_{\mathcal{Y}}$ for some $Y \in \mathcal{Y}$, then $f(1_Y) \in Ext_{\Lambda}^1(C,-)|_{\mathcal{Y}}$ correspondes to a short exact sequence $$0 \rightarrow Y \rightarrow X \overset{g}{\rightarrow} C \rightarrow 0.$$ How to get that the map $g$ is right minimal?