$\mathrm{Hom}(N,\bullet)$ is a functor from (($R$-alg)) to ((?))

Question

Let $R$ be a commutative ring with unity,

(($R$-alg)) the category of $R$-algebras,

and $N,M$ $R$-algebras.($N,M$ are commutative rings with unity and equipped with ring homomorphisms from $R$ mapping unity to unity.)

I am wondering $\mathrm{Hom}(N,\bullet)$ is a functor from (($R$-alg)) to what category?

I guess (($R$-mod)), the category of $R$-modules since as $N,M$ get $R$-module structures, $\mathrm{Hom}(N,M)$(which I guess the set of $R$-algebra homomorphisms from $N$ to $M$) is an $R$-module.

Question is from https://web.mit.edu/18.705/www/13Ed.pdf on page 44, Proposition(7.12). I am not sure if I understand the statement correctly.

Answer 1

The reason the hom functor is valued in sets, not in $R$-modules or even abelian groups, is that one can’t add and subtract homomorphisms of algebras.