Let $X$ be a compact complex manifold and let $Y \& Z$ be complex manifolds. Suppose $f: X \times Y \rightarrow Z$ is holomorphic and $f(X; y_0) = z_0$. Then, there is a holomorphic map $g: Y \rightarrow Z$ such that f factors
I tried it this way: From the above diagram it follows that, $g(y):=f(x_0,y)$
$f(x,y_0)=f(x',y_0) \hspace{1 cm} \forall x,x' \in X$
$\implies f(x,y)=f(x',y) \hspace{1 cm} \forall x,x' \in X, y \in Y$
for any open set $U \subset Z$, we have $\pi_y(f^{-1}(U^\complement))$ is a closed map.(since $X$ is compact, $\pi_y$ is a proper map \& proper maps are closed). So, g is continuous.
To prove holomorphic one need to choose coordinate chats for $X \times Y$ and charts for X which is induced by the map $\pi$.
But, my instructor said there is something wrong in the proof. I didn't get it.
Please help me out!