In a given category $\mathcal{C}$ I want to prove the following statement:
- If $U$ es a generator in the category $\mathcal{C}$ if and only if the left-exact functor $Hom_{\mathcal{C}}(U,-): \mathcal{C} \to Ab$ is and embedding where $Ab$ is the category of abelian groups.
Where I have the following definitions:
(a)An embedding is a faithful functor whichs sends different objects into different objects.\
(b) A generator in $\mathcal{C}$ is an object $U$ in $\mathcal{C}$ which for given two morphisms in $\mathcal{C}$ ; $f,g: A \to B$ such that $f \neq g$ then there exist a morphism $u: U \to A$ such $fu \neq gu$.
My idea to prove the statement is that for ($\Rightarrow$) since U is a generator we can easily see that the $Hom_{\mathcal{C}}(U,-)$ functor is faithfull since given $f,g$ such $f\neq g$ then $fu \neq gu$ bt how can I prove that the given functor sends different objects into different objects?? the other implication seems direct to prove. Anyways, thanks for your help.