Let $\Gamma$ be an Artin algebra with standard duality $D$. $I$ is the injective envelop of $\Gamma$ as $\Gamma$-module. Let $S$ be a set consisting of $\Gamma$-modules such that $X \in S$ iff there is an exact sequence $X \rightarrow B_0 \rightarrow B_1$ with $B_0,B_1 \in add (I)$. Then
$(1)$ How to show that $DHom_{\Gamma}(-,I): S \rightarrow mod End_{\Gamma}(I)$ is an equivalence? (I can not get the inverse)
$(2)$ If $\Gamma \in S$, how to get that $DHom_{\Gamma}(\Gamma,I)\cong DI$ is a generator and cogenerator of $modEnd_{\Gamma}(I)?$