What is $\mathrm{Hom}(\mathbb Q;\mathbb Q)$? Is it isomorphic to $\mathbb Q$? If yes, what is the isomorphism?
Is there a reference where the hom functor is explicitly studied with concrete examples like the above one?
What is $\mathrm{Hom}(\mathbb Q;\mathbb Q)$? Is it isomorphic to $\mathbb Q$? If yes, what is the isomorphism?
Is there a reference where the hom functor is explicitly studied with concrete examples like the above one?
So, you already know that you have at least one homomorphism for every rational number, given by "multiplication by $r$". You might also want to notice that composition of these homomorphisms coincides with multiplication of rationals; that is, you seem to have the multiplicative semigroup $\mathbb{Q}$ at the very least.
The question then is whether these are all the homomorphisms...
Suppose that $f\colon\mathbb{Q}\to\mathbb{Q}$ is an additive homomorphism, and $f(1)=a$. This clearly tells you what $f$ restricts to in the integer (namely, $f(m) = am$). Does it tell you anything else?
Well, what is $f(\frac{1}{2})$? Whatever it is, we have: $f(1) = f\left(\frac{1}{2}+\frac{1}{2}\right) = f\left(\frac{1}{2}\right)+f\left(\frac{1}{2}\right) = 2f\left(\frac{1}{2}\right).$ So in fact, we also know what the value of $f$ is at $\frac{1}{2}$.
Can we figure out how much $f$ is at other rationals, given how much it is at $1$? What does this tell us?