Injectivity, surjectivity and stability degree

Let $V$ be an FI-module over $k$ and let ${\mathrm{\Phi }}_a$ and $T$ be the coinvariant functors and transition maps of The coinvariant functors ${\mathrm{\Phi }}_a$ and the transition maps $T$. The injectivity degree $\mathrm{inj-deg}(V)$ is the least $s\ge 0$ such that for every $a\ge 0$ the transition map $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to {\mathrm{\Phi }}_a(V{)}_{n+1}$ is injective for all $n\ge s$; the surjectivity degree $\mathrm{surj-deg}(V)$ is the least $s\ge 0$ such that for every $a\ge 0$ the map $T$ is surjective for all $n\ge s$. The stability degree is

$\mathrm{stab-deg}(V):=\max (\mathrm{inj-deg}(V),\mathrm{surj-deg}(V))\mathrm{.}$

Each of the three is $+\mathrm{\infty }$ when no such $s$ exists; here the bounds are read as elements of the extended nonnegative integers $\mathbb{N}\cup \{+\mathrm{\infty }\}$ with the evident order, in which every nonempty subset of $\mathbb{N}$ has a least element (The natural numbers $\mathbb{N}$ (von Neumann)), so "the least $s$" is well defined. Equivalently, $T\text{\hspace{0.17em}⁣}:{\mathrm{\Phi }}_a(V{)}_n\to {\mathrm{\Phi }}_a(V{)}_{n+1}$ is an isomorphism for all $a\ge 0$ and all $n\ge \mathrm{stab-deg}(V)$: the map is an isomorphism exactly when it is both injective and surjective, and for finite bounds a threshold that controls all $a$ simultaneously is the larger of the two thresholds, while each of the two stated bounds remains the least threshold controlling its own property.

The three bounds are kept distinct: $\mathrm{stab-deg}(V)$ is the maximum of the other two, and nothing in the definition requires $\mathrm{inj-deg}(V)=\mathrm{surj-deg}(V)$. No finiteness is asserted: each bound is allowed to be $+\mathrm{\infty }$. No choice principle is used.