Let $X$ be a complex manifold. Then its tangent bundle $TX$ is equipped with the natural complex structure $J$ and thus is a smooth complex vector bundle. On the other hand, the holomorphic tangent bundle $T^{1,0}X \subset TX \otimes_{\mathbb{R}} \mathbb{C}$ is a holomorphic vector bundle on $X$. Then there is a natural isomorphism $TX \rightarrow T^{1,0}X, \ v \mapsto v-iJ(v)$ between the complex vector bundles. Does this mean that we should think of $TX$ as a holomorphic vector bundle (because the isomorphism is natural)?
I am confused because the (complex) basis $\{\frac{\partial }{\partial x_i}\}_i$ of $TX$ does not transform holomorphically, though I see that the basis $\{\frac{\partial }{\partial z_i}\}_i$ transforms holomorphically by holomorphic coordinate changes.