# Understanding a Map from rational points to the space of k-rational maps

I am currently working through this paper by Denef, and I have run into a little bit of a snafu with Lemma 3.1.

The setup is as follows: let $$K$$ be any field of characteristic zero, and let $$E_0$$ be the elliptic curve defined by the equation

$$Y^2 = X^3 + aX + b$$

Identify $$T$$ with the rational function $$(x,y) \to x$$ on $$E_0$$ and identify $$U$$ with the rational function $$(x,y) \to y$$ on $$E_0$$. Consider the field of rational functions $$F = K(T,U)$$, where $$U^2 = T^3 + aT + b$$. Denote the space of $$K$$-rational maps from $$E_0$$ to $$E_0$$ by $$Rat_K(E_0,E_0)$$ and let $$\psi_2$$ be the map

$$\psi_2: E_0(F) \to Rat_K(E_0,E_0)$$

which sends the point $$(V,W)$$ on $$E_0(F)$$ (the set of rational points of the elliptic curve $$Y^2 = X^3 + aX + b$$ where, in this case, $$(x,y)\in F$$) to the $$K$$-rational map

$$\psi_2(V,W): E_0(K) \to E_0(K): (x,y) \mapsto (V(x,y), W(x,y))$$

The claim is then that $$\psi_2$$ is a homomorphism. This seems clear to me since if we were to try and construct a rational map on $$E_0$$, this amount to sending a rational point $$(X,Y)$$ satisfying

$$Y^2 = X^3 + aX + b$$

to some other point satisfying the same elliptic curve equation, and we know that

$$W(x,y)^2 = V(x,y)^3 + aV(x,y) + b$$

since we made the assumption that these rational maps were members of $$F$$ (well, technically, they are identified with members of $$F$$ which are rational functions of $$T$$ and $$U$$, but I think that the point is clear).

My question is then how one would explicitly write down the homomorphism described by $$\psi_2$$? The first place that I am having trouble is trying to figure out how to write down the image of the identity of the elliptic curve would map to the identity map? I tried to write this down explicilty in terms of projective coordinates so that $$[T:U:Z] = [0:1:0] \mapsto Id_{E_0}$$, but I think that I am either missing or misunderstanding something since this is seeming to be a lot more trouble than I feel like it should be. Any help is greatly appreciated; thank you!