$$(f,g):\operatorname{Conf}^{\operatorname{sink}}_m(G,V) \rightarrow \operatorname{Conf}^{\operatorname{sink}}_n(G,V), (f,g)(x_1,\ldots,x_m)_i = \begin{cases} x_{f^{-1}(i)} & \text{ if $i \in f([m])$}\\ g(i) & \text{ otherwise.}\end{cases}$$ These assignments define a functor $Conf_{\bullet}(G,V)$ from $\operatorname{FI}_V$ to the category of topological spaces. For any fixed $q$, composition with the homology functor $H_q({\bullet})$ therefore defines an $\operatorname{FI}_V$-module over ${{\mathbb{Z}}}$ $$H_q(\operatorname{Conf}^{\operatorname{sink}}_{\bullet}(G,V)):\operatorname{FI}_V \rightarrow {{\mathbb{Z}}}\operatorname{-Mod}.$$\

We see that allowing collisions to happen at vertices enables us to introduce points into the sink configuration space in a continuous way. This ability seems to be quite rare in the study of configuration spaces in general. Notable cases in the classical setting where analogous maps exist are the configuration spaces of manifolds with boundary [@CEF][@MW], and manifolds with an everywhere non-vanishing vector field [@EW-G].\

One should note that given an injection $f:[n] \hookrightarrow [m]$, there is a forgetful map $f:\operatorname{Conf}^{\operatorname{sink}}_m(G,V) \rightarrow \operatorname{Conf}^{\operatorname{sink}}_n(G,V)$. Therefore the $\operatorname{FI}_V$-module $H_q(\operatorname{Conf}^{\operatorname{sink}}_{\bullet}(G,V))$ actually carries the structure of an $(\operatorname{FI}_V \times \operatorname{FI}^{\operatorname{op}})$-module. This extra structure imposes some fairly rigid restrictions on the module $H_q(\operatorname{Conf}^{\operatorname{sink}}_{\bullet}(G,V))$. For instance, it can be shown that this extra structure implies the dimension formula of Theorem \[sinkexpstab\] will hold for all $n$, instead of for $n$ sufficiently large. In the case of trees this follows from Theorem \[recursive\]. Showing that this 