I tried to do the exercise 1.1.5 on Liu's Algebraic Geometry and Arithmetic Curves. Namely,
Let $\rho:A\to B$ be a ring homomorphism, $M$ an $A$-module, and $N$ a $B$-module. How can I show that there exists a canonical isomorphism of $A$-modules $$\operatorname{Hom}_A(M,\rho_*N)\simeq \operatorname{Hom}_B(\rho^*M,N)?$$
Here $\rho_*$ is the module induced by natural way by $a\cdot y=\rho(a)y$ and $\rho^*M$ is the extension of scalars $M\otimes_AN$.
Do I have to use the universal property of tensor products?