Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and denote by $\bar{\mathbb{F}}_q$ its algebraic closure. Let $G$ be an algebraic group defined over $\bar{\mathbb{F}}_q$, and let $F$ be a Frobenius morphism which defines a $\mathbb{F}_q$-structure on $G$.
Suppose that $T$ is a $F$-stable maximal torus of $G$. It is known that the character $X(T)$ of $T$ is isomorphic to $\mathbb{Z}^n$ where $n$ is the rank of $T$. Thus the dual map $F^*$ acts linearly on $\mathbb{Z}^n$.
I have a question on a proof for the following statement.
$|T^F|=\mathrm{det}(F^*-1)$, where $|T^F|$ is the cardinality of $T^F$.
Proof: Consider the exact sequence $1\rightarrow T^F\xrightarrow{i} T\xrightarrow{F-1}T\rightarrow1$, and then $1\rightarrow X(T)\xrightarrow{F^*-1}X(T)\xrightarrow{p}\mathrm{Hom}(T^F,G_m)$. Because $|\mathrm{coker}(F^*-1)|=|\mathrm{det}(F^*-1)|$, it remains to show that $p$ is surjective.
The question is the following. In the note, it is shown that because $\mathrm{Hom}(\mathrm{Hom}(T^F,G_m),G_m)=T^F\rightarrow T=\mathrm{Hom}(X(T),G_m)$ is injective, the conclusion follows.
QUESTION: Does $G_m$ have any property in group category like injective module in module category? Or how can one obtain that $A\rightarrow B$ is surjective through the fact that $\mathrm{Hom}(B,G_m)\rightarrow\mathrm{Hom}(A,G_m)$ is injective?