Elliptic Curve and Weierstrass P function

There is a statement in Hartshorne's Algebraic Geometry that I do not see. It is the 3rd line in the proof of Proposition 4.18 of Chapter IV at page 329. I will summarize below:

Let P be the Weierstass P-function with periods 1 and tau which define a lattice M on C.

Let P' denote the derivative of P.

So, we can define the holomorphic function phi : C\M --> X by
phi(z) = (P(z), P'(z))
where X is an elliptic curve in the affine plane.

Now, let f : X --> X be a morphism of affine varieties. Then Hartshorne says the function
g : C --> C
(denoted as f bar in his book instead) induced by f via the pullback of phi must be holomorphic.

How does he conclude that?

This is how much I understand so far. Since f is an affine morphism, it is a polynomial map A^2 --> A^2. Hence, f is a polynomial map in terms of P and P'. Since P and P' generate the function field of elliptic functions, the composed map f o phi is a map of elliptic functions, which are holomorphic in C\M. Therefore, since
phi o g = f o phi,
we conclude that the composed map phi o g must be holomophic too. But I still can't conclude from here why the induced map g must be holomorphic.

Please help. Thanks.

--

-kira

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