Why are cones of left $\mathcal{X}$-approximations right $\mathcal{X}$-approximations?

Question

Let $A$ be a finite dimensional $k$-algebra, $D^b(A)$ be the bounded derived category of finite dimensional right $A$-modules. $\mathcal{X}$ is a subcategory of $D^b(A)$. Let $M\overset{f}{\rightarrow}N$ be a left $\mathcal{X}$-approximation of $M$. Is any of its homotopy cokernels a right $\mathcal{X}$-approximation of the cone of $f$?