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Small comment: certainly there's a natural map $\text{Hom}_R(X, R) \otimes_R S \to \text{Hom}_S(X, S)$ given by composing with $\varphi$ and extending by linearity. We can't hope for this map to be an isomorphism in general because the RHS sends colimits to limits but the LHS won't because if $\varphi$ is nontrivial then $S$ won't be flat over $R$. A typical example of what can go wrong here is to take $\varphi : \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z}$ and $X = \mathbb{Z}/2\mathbb{Z}^n$.