The simple regression model is
$y=\beta_0+\beta_1 x + u$,
where $Cov(x,u) \neq 0$. To obtain consistent estimators of $\beta_0$ and $\beta_1$ when $x$ and $u$ are correlated, we can use $z$ as an instrumental variable for $x$. The instrument has to satisfy the following two assumptions:
Wooldridge now writes: "instrument exogeneity means that $z$ should have no partial effect on $y$ (after x and ommited variables have been controlled for), and $z$ should be uncorrelated with the omitted variables."
I don't understand how the first assumption, $Cov(z,u)=0$, implies that $z$ has no partial effect on $y$, and why this is necessary. I understand that if $z$ were correlated with $u$ then $z$ would be endogenous and we couldn't obtain consistent estimators of $\beta_0$ and $\beta_1$. I just don't get why $z$ should have no partial effect on $y$ to be a valid instrument. Can someone explain this to me?