Let $\mathcal{C}$ be an additive category, $M$ an object in $\mathcal{C}$, and $add(M)$ the full subcategory of $\mathcal{C}$ consisting of all direct summands of finite sums of copies of $M$. Suppose $X\rightarrow M_n \rightarrow \cdots \rightarrow M_2 \rightarrow M_1 \rightarrow Y$ is a (not necessarily exact) sequence of morphisms in $\mathcal{}$ with $M_i \in \textrm{add}(M)$.
Let $\Lambda$ be the endomorphism ring of $V$, where $V := X\oplus M$. And we have an exact sequence of $\Lambda$-modules: \begin{align*} 0 \rightarrow \textrm{Hom}_{\mathcal{C}}(V,X) \rightarrow \textrm{Hom}_{\mathcal{C}}(V,M_n) \rightarrow \cdots \rightarrow \textrm{Hom}_{\mathcal{C}}(V,M_2) \rightarrow \textrm{Hom}_{\mathcal{C}}(V,M_1 \oplus M) \rightarrow T \rightarrow 0. \end{align*} In a paper, he says that "Applying $\textrm{Hom}_{\Lambda}(-,\textrm{Hom}_{\mathcal{C}}(V ,M))$ to this sequence, we get a sequence which is isomorphic to the following sequence \begin{align*} 0 \rightarrow \textrm{Hom}_{\Lambda}(T,\textrm{Hom}_{\mathcal{C}}(V,M)) \rightarrow \textrm{Hom}_{\mathcal{C}}(M_1 \oplus M,M) \rightarrow \textrm{Hom}_{\mathcal{C}}(M_2,M) \rightarrow \cdots \rightarrow \textrm{Hom}_{\mathcal{C}}(M_n,M) \rightarrow \textrm{Hom}_{\mathcal{C}}(X,M) \rightarrow 0 \end{align*} So is there anyone tell me how to apply this functor $\textrm{Hom}_{\Lambda}(-,\textrm{Hom}_{\mathcal{C}}(V ,M))$ to get that sequence? Whether $\text{Hom}_{\Lambda}(\text{Hom}_{\mathcal{C}}(V,M_1), \text{Hom}_{\mathcal{C}}(V,M_2)) \cong \text{Hom}_{\mathcal{C}}(M_1,M_2)$?
you can see lemma 3.1 in page 202 of "Elements of the Representation Theory of Associative Algebras 1". Note that $M_i\in\text{add}(M)\subset\text{add}(V).$