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If your categories are small and Cauchy-complete, a natural choice is to take the presheaf construction. (Cauchy-completeness is necessary because $F$ is only recoverable up to Cauchy completion.)
Perhaps it would be worth contacting Robin Houston? I notice another result about monoidal categories is attributed to him on the nLab page for semicartesian categories, but without a proof. Perhaps he could be persuaded to write these up.
