= (f \circ f',h),$$ where $$h(x) = \begin{cases} g(x) &\text{ if $x \notin \text{im}f$}\\ g'(f^{-1}(x)) &\text{ otherwise.} \end{cases}$$ Note that if $|V| = d$, then $\operatorname{FI}_V$ is equivalent to the category $\operatorname{FI}_d$, which can be seen discussed in [@R2], [@SS], [@SS2], [@Sa1] and [@Sa2]. The work [@SS2] approaches this category from a very different, but equivalent, perspective. Given a commutative ring $k$, an **$\operatorname{FI}_V$-module over $k$** is a functor $W:\operatorname{FI}_V \rightarrow k\operatorname{-Mod}$. A **morphism of $\operatorname{FI}_V$** modules is a natural transformation of functors. The category of $\operatorname{FI}_V$-modules is denoted $\operatorname{FI}_V\operatorname{-Mod}$. This is an abelian category, with abelian operations defined point-wise.\
We will often use $W_S$ to denote $W(S)$, and $(f,g)_{\text{*}}$ to denote $W(f,g)$. Given a finite set $S$, we define the **free $\operatorname{FI}_V$-module on $S$** to be the unique module $M(S)$ which satisfies $$\operatorname{Hom}_{\operatorname{FI}_V\operatorname{-Mod}}(M(S),W) = W_S$$ for every $\operatorname{FI}_V$-module $W$. Equivalently, $M(S)$ is the $\operatorname{FI}_V$-module which is defined on points by $$M(S)_T = k[\operatorname{Hom}_{\operatorname{FI}_V}(S,T)],$$ and for which morphisms act by composition. We say that an $\operatorname{FI}_V$-module $W$ is **finitely generated** if it is surjected onto by a direct sum of free modules. That is, if there is a finite collection $\{v_i\} \subseteq \sqcup_{S} W_S$ which generates the whole of $W$.\

