Let $k$ be a field and $\mathcal{C}$ be a dg-category over $k$. It is standard to define the homotopy category $H^0(\mathcal{C})$ as the category consisting the same objects as $\mathcal{C}$ but morphisms between two objects $x$ and $y$ are defined as $$ H^0(\mathcal{C})(x,y):=Z^0(\mathcal{C})(x,y)/B^0(\mathcal{C})(x,y). $$

We could also consider the category $Z^0(\mathcal{C})$ and the class $W$ of homotopic invertible morphisms in $Z^0(\mathcal{C})$. More precisely, $$ W=\{f\in Z^0(\mathcal{C})(x,y) \text{ for some }x \text{ and }y~|~f \text{ is invertible in } H^0(\mathcal{C})(x,y)\}. $$

We could localize the category $Z^0(\mathcal{C})$ by formally inverting $W$ and obtain a category $W^{-1}Z^0(\mathcal{C})$. There is a natural functor $W^{-1}Z^0(\mathcal{C})\to H^0(\mathcal{C})$.

My question is: Is $W^{-1}Z^0(\mathcal{C})$ equivalent to $H^0(\mathcal{C})$?