Let $R\to R'$ be a morphism of commutative rings and $M$, $N$ two finitely generated projective $R$-modules. Is it true that the map \begin{align*} \text{Hom}_R(M,N) \otimes R'&\to \text{Hom}_{R'}(M\otimes R',N\otimes R')\\ (f\otimes r'&\mapsto ((m\otimes s')\mapsto (r'f(m)\otimes s')) \end{align*} is surjective?
This is true if $R\to R'$ is a localization, see e.g. Show that this mapping between localized modules is an isomorphism. This, however, uses exactness of localization, and $-\otimes R'$ is only right exact.