Reading Griffiths and Harris, it is stated as an "obvious" fact that for a tensor product $E \otimes E'$ of holomorphic vector bundles, $(D_E \otimes 1 + 1 \otimes D_{E'})'' = \bar \partial$ where $D_E$ is the unique connection compatible with the complex structure of $E$ (so its complex component $D_E''$ is equal to $\bar \partial$).
However, I don't really understand how $\bar \partial$ is being applied to the tensor product of vector bundles.
A couple pages back it is explained how if we choose a local holomorphic frame we can essentially try to apply $\bar \partial$ "normally" (as if we were in $\mathbb{C}^n$), and in the case we are looking at (applying it to zero forms) I think it should locally look like $\sum_i d\bar z_i \frac{\partial}{\partial \bar z_i}$ but how do we get from here to the description in terms of tensor products, and from that to $(D_E \otimes 1) + 1 \otimes D_{E'})'' = \bar \partial$?
Clearly I am very confused so any better explanation of this would help.