The following question arises from the proof of "bend-and-break" lemma:

Let $X$ be a projective variety over $\mathbb{C}$ and $C$ be an irrational smooth curve. Let $c \in C$ be a fixed closed point. Let $f: C \to X$ be a nonconstant morphism such that $f(c)=x$.

Suppose there exists an irreducible one dimensional variety $T \subset Mor(C,X;f|_{c})$ passing through $[f]$ (we use $[f]$ to denote the point corresponding to $f$ in the moduli space), where by $Mor(C,X;f|_{c})$, we mean the moduli space of morphisms from $C$ to $X$ such that any morphism maps the point $c$ to $x$. Let $e$ be the evaluation map restricted to $C \times T$, that is

$$e: C \times T \to C \times Mor(C,X;f|_{c}) \to X.$$

My questions is, why $\dim(e(C \times T)) >1$?

I understand when $g(C)>0$, with one point fixed, $C$ only has finite automorphism. But I don't know how to use this fact to show the claim.