Let $\varphi: \mathbb{Q}[X] \longrightarrow R$ an inclusion of commutative rings. Suppose that the map $$- \circ \varphi: \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(R, R^{\otimes_{\mathbb{Q}} n}) \longrightarrow \operatorname{Hom}_{\mathbb{Q}\operatorname{-alg}}(\mathbb{Q}[X], R^{\otimes_{\mathbb{Q}} n}) \cong R^{\otimes_{\mathbb{Q}} n}$$ is a bijection for all nonnegative integers $n$ (where $R^{\otimes_{\mathbb{Q}} 0} = \mathbb{Q}$). Must the map $\varphi$ be an isomorphism? If so, then, replacing $\mathbb{Q}$ with an arbitrary field $k$ of characteristic zero, is the corresponding statement true?

The question is motivated by questions about plethories, specifically, Classification of plethories over $\mathbb{Q}$

In the original version of the question I did not include the assumption that $\operatorname{char} k = 0$. If you leave out that assumption then $\varphi$ need not be an isomorphism. A nice counterexample is provided below by Julian Rosen.