Why does parameterizing a curve with its length yield $|g(T,T)|=1$?

Question

Let $l(t)=\int^t_{t_0}|T|(t')dt'$, where $T$ is a tangent vector to some curve $C(t)$.

Why does setting this function as a parameterization of the curve $C$, hence letting $l(t)=\psi C(t)$, imply $|T|^2=|g(T,T)|=1$? ($g$ here is the metric tensor)I can't seem to figure it out.

Thanks.