I would like to ask about terminology for a partition of unity for an étale groupoid.

I am reading the lecture notes "Cohomology of Stacks" by Prof. Behrend. A partition of unity is defined in Definition 22. Let $X_1 \rightrightarrows X_0$ be a Lie groupoid satisfying the following conditions.

1) The source map $s$ and the target map $t$ are both étale (i.e., induce isomorphisms on tangent spaces).

2) The map $(s,t):X_1 \to X \times X$ is proper and unramified, with finite fibers (unramified means injective on tangent spaces).

For such a Lie groupoid, a partition of unity is defined as follows.

A partition of unity for the groupoid $X_1 \rightrightarrows X_0$ is an $\mathbb{R}$-valued $C^\infty$-function $\rho$ on $X_0$ with the property that $s^*\rho$ has proper support with respect to $t:X_1 \to X_0$ and $t_!s^* \rho \equiv 1$.

a) What does "$\rho$ has proper support with respect to $t:X_1 \to X_0$" means?

b) How do we define the operator $t_!$?

I would be most grateful if you could tell me references on these terminology.