The proof of this lemma is the same as the proof of Lemma \ref{lemma-from-weil-to-functor-classical}; we urge the reader to read the proof of that lemma instead. \medskip\noindent By Lemma \ref{lemma-characterize-motives} it suffices to construct a functor $G$ on the category of smooth projective schemes over $k$ with morphisms given by correspondences of degree $0$ such that the image of $G(c_2)$ on $G(\mathbf{P}^1_k)$ is an invertible graded $F$-vector space. \medskip\noindent Let $X$ be a smooth projective scheme over $k$. There is a canonical decomposition $$ X = \coprod\nolimits_{0 \leq d \le \dim(X)} X_d $$ into open and closed subschemes such that $X_d$ is equidimensional of dimension $d$. By Lemma \ref{lemma-weil-additive} we have correspondingly $$ H^*(X) \longrightarrow \prod\nolimits_{0 \leq d \le \dim(X)} H^*(X_d) $$ If $Y$ is a second smooth projective scheme over $k$ and we similarly decompose $Y = \coprod Y_e$, then $$ \text{Corr}^0(X, Y) = \bigoplus \text{Corr}^0(X_d, Y_e) $$ As well we have $X \otimes Y = \coprod X_d \otimes Y_e$ in the category of correspondences. From these observations it follows that it suffices to construct $G$ on the category whose objects are equidimensional smooth projective schemes over $k$ and whose morphisms are correspondences of degree $0$. (Some details omitted.) \medskip\noindent Given an equdimensional smooth projective scheme $X$ over $k$ we set $G(X) = H^*(X)$. Observe that $G(X) = 0$ if $X = \emptyset$ (Lemma \ref{lemma-unit}). Thus maps from and to $G(\emptyset)$ are zero and we may and do assume our schemes are nonempty in the discussions below. \medskip\noindent Given a correspondence $c \in \text{Corr}^0(X, Y)$ between nonempty equidmensional smooth projective schemes over $k$ we consider the map $G(c) : G(X) = H^*(X) \to G(Y) = H^*(Y)$ given by the rule $$ a \longmapsto G(c)(a) = \text{pr}_{2, *}(\gamma(c) \cup \text{pr}_1^*a) $$ It is clear that $G(c)$ is additive in $c$ and hence $\mathbf{Q}$-linear. Compatibility of $\gamma$ with pullbacks, pushforwards, and intersection products given by axioms (C)(a), (C)(b), and (C)(c) shows that we have $G(c' \circ c) = G(c') \circ G(c)$ if $c' \in \text{Corr}^0(Y, Z)$. Namely, for $a \in H^*(X)$ we have \begin{align*} (G(c') \circ G(c))(a) & = \text{pr}^{23}_{3, *}(\gamma(c') \cup \text{pr}^{23, *}_2(\text{pr}^{12}_{2, *}(\gamma(c) \cup \text{pr}^{12, *}_1a))) \\ & = \text{pr}^{23}_{3, *}(\gamma(c') \cup \text{pr}^{123}_{23, *}(\text{pr}^{123, *}_{12}(\gamma(c) \cup \text{pr}^{12, *}_1 a))) \\ & = \text{pr}^{23}_{3, *} \text{pr}^{123}_{23, *}( \text{pr}^{123, *}_{23}\gamma(c') \cup \text{pr}^{123, *}_{12}\gamma(c) \cup \text{pr}^{123, *}_1 a) \\ & = \text{pr}^{23}_{3, *} \text{pr}^{123}_{23, *}( \gamma(\text{pr}^{123, *}_{23}c') \cup \gamma(\text{pr}^{123, *}_{12}c) \cup \text{pr}^{123, *}_1 a) \\ & = \text{pr}^{13}_{3, *} \text{pr}^{123}_{13, *}( \gamma(\text{pr}^{123, *}_{23}c' \cdot \text{pr}^{123, *}_{12}c) \cup \text{pr}^{123, *}_1 a) \\ & = \text{pr}^{13}_{3, *}( \gamma(\text{pr}^{123}_{13, *}(\text{pr}^{123, *}_{23}c' \cdot \text{pr}^{123, *}_{12}c)) \cup \text{pr}^{13, *}_1 a) \\ & = G(c' \circ c)(a) \end{align*} with obvious notation. The first equality follows from the definitions. The second equality holds because $\text{pr}^{23, *}_2 \circ \text{pr}^{12}_{2, *} = \text{pr}^{123}_{23, *} \circ \text{pr}^{123, *}_{12}$ as follows immediately from the description of pushforward along projections given in Lemma \ref{lemma-pr2star}. The third equality holds by Lemma \ref{lemma-pushforward} and the fact that $H^*$ is a functor. The fourth equalith holds by axiom (C)(a) and the fact that the gysin map agrees with flat pullback for flat morphisms (Chow Homology, Lemma \ref{chow-lemma-lci-gysin-flat}). The fifth equality uses axiom (C)(c) as well as Lemma \ref{lemma-pushforward} to see that $\text{pr}^{23}_{3, *} \circ \text{pr}^{123}_{23, *} = \text{pr}^{13}_{3, *} \circ \text{pr}^{123}_{13, *}$. The sixth equality uses the projection formula from Lemma \ref{lemma-pushforward} as well as axiom (C)(b) to see that $ \text{pr}^{123}_{13, *} \gamma(\text{pr}^{123, *}_{23}c' \cdot \text{pr}^{123, *}_{12}c) = \gamma(\text{pr}^{123}_{13, *}( \text{pr}^{123, *}_{23}c' \cdot \text{pr}^{123, *}_{12}c))$. Finally, the last equality is the definition. \medskip\noindent To finish the proof that $G$ is a functor, we have to show identities are preserved. In other words, if $1 = [\Delta] \in \text{Corr}^0(X, X)$ is the identity in the category of correspondences (Lemma \ref{lemma-category-correspondences}), then we have to show that $G([\Delta]) = \text{id}$. This follows from the determination of $\gamma([\Delta])$ in Lemma \ref{lemma-class-diagonal} and Lemma \ref{lemma-pr2star}. This finishes the construction of $G$ as a functor on smooth projective schemes over $k$ and correspondences of degree $0$. \medskip\noindent By Lemma \ref{lemma-base} we have that $G(\Spec(k)) = H^*(\Spec(k))$ is canonically isomorphic to $F$ as an $F$-algebra. The K\"unneth axiom (B)(a) shows our functor is compatible with tensor products. Thus our functor is a functor of symmetric monoidal categories. \medskip\noindent We still have to check that the image of $G(c_2)$ on $G(\mathbf{P}^1_k) = H^*(\mathbf{P}^1_k)$ is an invertible graded $F$-vector space (in particular we don't know yet that $G$ extends to $M_k$). By Lemma \ref{lemma-cohomology-P1} we only have nonzero cohomology in degrees $0$ and $2$ both of dimension $1$. We have $1 = c_0 + c_2$ is a decomposition of the identity into a sum of orthogonal idempotents in $\text{Corr}^0(\mathbf{P}^1_k, \mathbf{P}^1_k)$, see Example \ref{example-decompose-P1}. Further we have $c_0 = a \circ b$ where $a \in \text{Corr}^0(\Spec(k), \mathbf{P}^1_k)$ and $b \in \text{Corr}^0(\mathbf{P}^1_k, \Spec(k))$ and where $b \circ a = 1$ in $\text{Corr}^0(\Spec(k), \Spec(k))$, see proof of Lemma \ref{lemma-inverse-h2}. Thus $G(c_0)$ is the projector onto the degree $0$ part. It follows that $G(c_2)$ must be the projector onto the degree $2$ part and the proof is complete.