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Namely, for every functor $F$, is $(F, (A_0, A_1)\mapsto[F(\iota_0(A_0, A_1)), F(\iota_1(A_0, A_1))])$ monoidal? (For reference, $\iota_i(A_0, A_1):A_i\to A_0+A_1$ is an injection of the categorical sum $A_0+A_1$.)

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