Proposition 45.9.12. Let $k$ be a field. Let $F$ be a field of characteristic $0$. There is a $1$-to-$1$ correspondence between the following

1. data (D0), (D1), (D2), and (D3) satisfying (A), (B), and(C), and

2. $\mathbf{Q}$-linear symmetric monoidal functors

$G : M_ k \longrightarrow \text{graded }F\text{-vector spaces}$

such that $G(\mathbf{1}(1))$ is nonzero only in degree $-2$.

Proof. Given $G$ as in (2) by setting $H^*(X) = G(h(X))$ we obtain data (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C), see Lemma 45.9.10 and its proof.

Conversely, given data (D0), (D1), (D2), and (D3) satisfying (A), (B), and (C) we get a functor $G$ as in (2) by the construction of the proof of Lemma 45.9.11.

We omit the detailed proof that these constructions are inverse to each other. $\square$

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