What does it mean for a pivotal quantity to be monotone in $\theta$?

Question

I'm told that a pivotal quantity is a function $g(\mathbf X,\theta)$ of the data $\mathbf X$ and the parameter $\theta$, satisfying

1. The distribution of $g(\mathbf X,\theta)$ is known and independent of $\theta$,
2. For each fixed $\mathbf X$, $g(\mathbf X,\theta)$ is a monotonic function of $\theta$.

I do not understand the 2nd criterion. Why does $\mathbf X$ need to be fixed? What does it mean for $g(\mathbf X,\theta)$ to be a monotonic function of $\theta$? Why does it need to be monotonic?