Property of the syzygy functor of $\operatorname{\underline{mod}} A$

Question

Let $A$ be an artin algebra. We denote by $\operatorname{mod}A$ the category of finitely generated left $A$-modules. We denote by $[A]$ the ideal of morphisms between $A$-modules which factor through a projective $A$-module. Let $\operatorname{\underline{mod}} A := (\operatorname{mod} A)/[A]$ and $\operatorname{\underline{Hom}}_A(X,Y)$ the set of morphisms in $\operatorname{\underline{mod}}A$.For any $X \in\operatorname{mod} A$, there is a projective cover $f:P(X) \rightarrow X $, we denote the kernel of $f$ by $\Omega X$. Then we have the syzygy functor $\Omega : \operatorname{\underline{mod}} A \rightarrow \operatorname{\underline{mod}}A $.

1. Set $\mathcal{C}= \{ M \in \operatorname{mod}A \mid \operatorname{Ext}^1_A(M,A)=0 \}$. How to check that $\Omega : \operatorname{\underline{Hom}}_A(X,Y)\rightarrow \operatorname{\underline{Hom}}_A(\Omega X, \Omega Y)$ is bijective for $X \in \mathcal{C}$ and $Y \in \operatorname{mod}A$?
2. Is there any result or conclusion about that under some conditions, $\Omega : \operatorname{\underline{Hom}}_A(X,Y)\rightarrow \operatorname{\underline{Hom}}_A (\Omega X, \Omega Y)$ is bijective for $X,Y \in \operatorname{mod}A$?