Edward Miller is a pretty decent person.
Earlier, when getting ready to enter the Misty Sea, he offered Carson Flores a Detoxification Pill. At that time, Edward didn't know the Detoxification Pill he had prepared couldn't handle the miasma, which shows that he didn't see Carson as a competitor.
Perhaps it was that battle that made him feel a bond with Carson, thus considering him a partner on this adventure.
For this reason, even if Edward didn't team up with Carson, but acted on his own, using their respective strategies to compete for the Rainbow Jade Rhinoceros Horn, Carson would still gladly offer the Detoxification Pill.
Edward's actions were quite magnanimous, ignoring the difference in strength between himself and Carson, and directly offering a promise of equal partnership.
"How long can the effect of this Detoxification Pill last?"
