Let $\mathcal{C}$ be an additive category, $\Psi (C)$ the category of all complexes over $\mathcal{C}$ with chain maps, $\mathcal{K} (C)$ the homotopy category of $\Psi (C)$, the derived category of $\mathcal{C}$ denoted by $\mathcal{D} (C)$, $ \mathcal{D}(C)$ is the localization of $\mathcal{K} (C)$ at all quasi-isomorphisms.
Suppose $X,M,Y$ are objects in $\mathcal{C}$ $\mathcal{}$, we denote by $add(M)$ the full subcategory of $\mathcal{C}$ consisting of all direct summands of direct sums of finitely many copies of $M$. Let $M_{0} \in add(M)$, $ X \rightarrow M_0 \rightarrow Y$ be an $add(M)$-exact sequence in $\mathcal{C}$ .
Supose $V :=X \oplus M$, $\Lambda :=End_{\mathcal{C}} (V)$ Moreover, there exists the following exact sequence of $\Lambda$-modules: $0 \rightarrow Hom_{\mathcal{C}} (V,X) \stackrel{f}{\longrightarrow} Hom_{\mathcal{C}} (V,M_0) \stackrel{g}{\longrightarrow} Hom_{\mathcal{C}}(V,Y)$. Let $T :=Hom_{\mathcal{C}}(V,M) \oplus Im(g)$. Then we can get another exact sequence of $\Lambda$-modules: $0 \rightarrow Hom_{\mathcal{C}} (V,X) \rightarrow Hom_{\mathcal{C}} (V,M \oplus M_0) \rightarrow T \rightarrow 0$.
Let $P^{\bullet}$ be the following complex: $0 \rightarrow Hom_{\mathcal{C}} (V,X) \rightarrow Hom_{\mathcal{C}} (V,M \oplus M_0)\rightarrow 0$. Since $X \in add(V)$ and $M_0 \in add(M) \subseteq add(V)$, we can get both $Hom_{\mathcal{C}}(V,X)$ and $Hom_{\mathcal{C}}(V,M \oplus M_0)$ are finitely generated projective $\Lambda$-modules.
I want to ask how to use that $Hom_{\mathcal{C}}(V,X)$ and $Hom_{\mathcal{C}}(V,M \oplus M_0)$ are finitely generated projective $\Lambda$-modules to get the following isomorphism of rings $End_{\Lambda}(T) \cong End_{\mathcal{D}(\Lambda)}(P^{\bullet}) \cong End_{\mathcal{K}(\Lambda)}(P^{\bullet})$?