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 $G(\mathbf{P}^1)$ is an invertible graded $F$-vector space. Since every smooth projective scheme is canonically a disjoint union of smooth projective varieties, it suffices to construct $G$ on the category whose objects are smooth projective varieties and whose morphisms are correspondences of degree $0$. (Some details omitted.) \medskip\noindent Given a smooth projective variety $X$ we set $G(X) = H^*(X)$. \medskip\noindent Given a correspondence $c \in \text{Corr}^0(X, Y)$ between smooth projective varieties 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-classical}. The third equality holds by Lemma \ref{lemma-pushforward-classical} 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-classical} 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-classical} 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 (see Lemma \ref{lemma-category-correspondences} and its proof), then we have to show that $G([\Delta]) = \text{id}$. This follows from the determination of $\gamma([\Delta])$ in Lemma \ref{lemma-class-diagonal-classical} and Lemma \ref{lemma-pr2star-classical}. This finishes the construction of $G$ as a functor on smooth projective varieties and correspondences of degree $0$. \medskip\noindent It follows from axioms (A)(c) and (A)(d) that $G(\Spec(k)) = H^*(\Spec(k))$ is canonically isomorphic to $F$ as an $F$-algebra. The K\"unneth axiom (B) 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 $G(\mathbf{P}^1)$ is an invertible graded $F$-vector space (in particular we don't know yet that $G$ extends to $M_k$). By axiom (A)(d) the map $\int_{\mathbf{P}^1} : H^2(\mathbf{P}^1) \to F$ is an isomorphism. By axiom (A)(b) we see that $\dim_F H^0(\mathbf{P}^1) = 1$. By Lemma \ref{lemma-square-diagonal-classical} and axiom (A)(c) we obtain $2 - \dim_F H^1(\mathbf{P}^1) = c_1(T_{\mathbf{P}^1}) = 2$. Hence $H^1(\mathbf{P}^1) = 0$. Thus $$ G(\mathbf{P}^1) = H^0(\mathbf{P}^1) \oplus H^2(\mathbf{P}^1) $$ Recall that $1 = c_0 + c_2$ is a decomposition of the identity into a sum of orthogonal idempotents in $\text{Corr}^0(\mathbf{P}^1, \mathbf{P}^1)$, see Example \ref{example-decompose-P1}. We have $c_0 = a \circ b$ where $a \in \text{Corr}^0(\Spec(k), \mathbf{P}^1)$ and $b \in \text{Corr}^0(\mathbf{P}^1, \Spec(k))$ and where $b \circ a = 1$ in $\text{Corr}^0(\Spec(k), \Spec(k))$, see proof of Lemma \ref{lemma-inverse-h2}. Since $F = G(\Spec(k))$, it follows from functoriality that $G(c_0)$ is the projector onto the summand $H^0(\mathbf{P}^1) \subset G(\mathbf{P}^1)$. Hence $G(c_2)$ must necessarily be the projection onto $H^2(\mathbf{P}^1)$ and the proof is complete.