Proof.
Denote U in \mathcal{C} the invertible object which is assumed to exist in the statement of the lemma. We extend F to motives by setting
F(X, p, m) = \left(\text{the image of the projector }F(p)\text{ in }F(X)\right) \otimes U^{\otimes -m}
which makes sense because U is invertible and because \mathcal{C} is Karoubian. An important feature of this choice is that F(X, c_2, 0) = U. Observe that
\begin{align*} F((X, p, m) \otimes (Y, q, n)) & = F(X \times Y, p \otimes q, m + n) \\ & = \left(\text{the image of }F(p \otimes q)\text{ in }F(X \times Y)\right) \otimes U^{\otimes -m - n} \\ & = F(X, p, m) \otimes F(Y, q, n) \end{align*}
Thus we see that our rule is compatible with tensor products on the level of objects (details omitted).
Next, we extend F to morphisms of motives. Suppose that
a \in \mathop{\mathrm{Hom}}\nolimits ((Y, p, m), (Z, q, n)) = q \circ \text{Corr}^{n - m}(Y, Z) \circ p \subset \text{Corr}^{n - m}(Y, Z)
is a morphism. If n = m, then a is a correspondence of degree 0 and we can use F(a) : F(Y) \to F(Z) to get the desired map F(Y, p, m) \to F(Z, q, n). If n < m we get canonical identifications
\begin{align*} s : F((Z, q, n)) & \to F(Z, q, m) \otimes U^{m - n} \\ & \to F(Z, q, m) \otimes F(X, c_2, 0) \otimes \ldots \otimes F(X, c_2, 0) \\ & \to F((Z, q, m) \otimes (X, c_2, 0) \otimes \ldots \otimes (X, c_2, 0)) \\ & \to F((Z \times X^{m - n}, q \otimes c_2 \otimes \ldots \otimes c_2, m)) \end{align*}
Namely, for the first isomorphism we use the definition of F on motives above. For the second, we use the choice of U. For the third we use the compatibility of F on tensor products of motives. The fourth is the definition of tensor products on motives. On the other hand, since we similarly have an isomorphism
\sigma : (Z, q, n) \to (Z \times X^{m - n}, q \otimes c_2 \otimes \ldots \otimes c_2, m)
(see proof of Lemma 45.4.6). Composing a with this isomorphism gives
\sigma \circ a \in \mathop{\mathrm{Hom}}\nolimits ((Y, p, m), (Z \times X^{m - n}, q \otimes c_2 \otimes \ldots \otimes c_2, m))
Putting everything together we obtain
s^{-1} \circ F(\sigma \circ a) : F(Y, p, m) \to F(Z, q, n)
If n > m we similarly define isomorphisms
t : F((Y, p, m)) \to F((Y \times X^{n - m}, p \otimes c_2 \otimes \ldots \otimes c_2, n))
and
\tau : (Y, p, m)) \to (Y \times X^{n - m}, p \otimes c_2 \otimes \ldots \otimes c_2, n)
and we set F(a) = F(a \circ \tau ^{-1}) \circ t. We omit the verification that this construction defines a functor of symmetric monoidal categories.
\square
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