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 $\mathbb{Q}$](https://mathoverflow.net/questions/113989/classification-of-plethories-over-mathbbq)


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. 



Characterizing $\mathbb{Q}[X]$ via a property of its tensor powers
To answer my own question...I just proved (finally) that it's true assuming a positive answer to the question [Classification of plethories over $\mathbb{Q}$](https://mathoverflow.net/questions/113989/classification-of-plethories-over-mathbbq), that is, assuming that every $\mathbb{Q}$-plethory is linear. Basically, the proof goes by showing that the given condition implies that $R$ has a $\mathbb{Q}$-plethory structure such that the comonad structure $W\_R \longrightarrow W\_R W\_R$ is an isomorphism, and then its purported linearity implies that $R$ is isomorphic to $\mathbb{Q}[X]$ as a $\mathbb{Q}$-plethory.


