Let $\mathcal{A}$ be an abelian category and $\mathcal{K}(\mathcal{A})$ be the homotopy category of chain complexes in $\mathcal{A}$. It is well-known that $\mathcal{K}(\mathcal{A})$ is idempotent complete in the following sense: Let $X$ be an object in $\mathcal{K}(\mathcal{A})$ and $\alpha: X\rightarrow X$ be a morphism which satisfies $$ \alpha^2=\alpha \text{ in } \mathcal{K}(\mathcal{A}), $$ then there exist $Y$ in $\mathcal{K}(\mathcal{A})$ together with $i: Y\rightarrow X$ and $X\rightarrow Y$ such that $$ pi=id_Y \text{ and } ip=\alpha. $$

Now since $X$ is a chain complex and $\alpha$ is a chain map, we can form the image of $\alpha$, $\alpha(X)$ which is a subcomplex of $X$. $\textbf{My question}$ is: do we have an isomorphism $$ Y\cong \alpha(X) \text{ in } \mathcal{K}(\mathcal{A})? $$

Let $\mathcal{A}$ be an abelian category and $\mathcal{K}(\mathcal{A})$ be the homotopy category of chain complexes in $\mathcal{A}$. It is well-known that $\mathcal{K}(\mathcal{A})$ is idempotent complete in the following sense: Let $X$ be an object in $\mathcal{K}(\mathcal{A})$ and $\alpha: X\rightarrow X$ be a morphism which satisfies $$ \alpha^2=\alpha \text{ in } \mathcal{K}(\mathcal{A}), $$ then there exist $Y$ in $\mathcal{K}(\mathcal{A})$ together with $i: Y\rightarrow X$ and $p: X\rightarrow Y$ such that $$ pi=id_Y \text{ and } ip=\alpha. $$

Now since $X$ is a chain complex and $\alpha$ is a chain map, we can form the image of $\alpha$, $\alpha(X)$ which is a subcomplex of $X$. $\textbf{My question}$ is: do we have an isomorphism $$ Y\cong \alpha(X) \text{ in } \mathcal{K}(\mathcal{A})? $$