Property of Complete Variety

I have a question about a step in the proof from Lang’s “Abelian Varieties” (page 20):

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By definition an abelian variety $ A$ over field $ k$ is a proper smooth $ k$ -group scheme that is irreducible.

In the Theorem 1 where we have to show that an abelian variety is commutative the author says in the red tagged line that it suffice to show that

$ $ \dim T \le \dim A$ $

where $ T$ is the locus if $ (x, yxy^{-1}) \in A \times A$ .

Following step isn’t clear:

Why completeness of $ A$ implies that the point $ (e,a) \in T \cap (e \times A)$ has a preimage $ (e,b)$ in $ A \times A$ under the map $ (x,y) \mapsto (x, yxy^{-1})$ ?

I’m working with wiki’s definition of completeness.