Caicedo was heavily linked with a move away from the club in the January transfer window, with the Gunners and Chelsea both interested.
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The Ecuadorian international even made a plea on social media to leave towards the end of the window, much to the dismay of fans, but Barber insists there are no hard feelings between player and club.
Speaking on Drive, he said: "There was never a moment in this window where we felt we needed or wanted to sell Moises.
"He's only 20 years of age, he needed an arm around the shoulder, he needed a bit of support from everybody. We know that his head was turned, we understand why, but we also felt he had a job to do here and he was on a good contract, a long contract.
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"We've since improved that. Moises to his absolute credit has got his head down, he's worked really hard, he's positive, he's happy around the club again.
"He's very, very popular and is a very, very good footballer so at the moment we're focussed on what we can do in the remaining games of this season and Moises is a big part of that."
Brighton currently sit seventh in the Premier League, and much of their success so far is down to the playing talent they have at their disposal.
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The club has been hailed for their scouting and recruitment, and Barber is aware of how effective their strategy has been.
Barber said: "We're very confident in our recruitment processes. We've had the same processes in place now since Chris Hughton's time here and even before that.
"Each year that goes by, we refine those processes, we look and fish in different ponds to some of the bigger clubs. We can't afford to compete with the traditionally big six or seven clubs in this country so we have to do it differently.
"We also have to use our academy, we have to bring through players like Evan Ferguson that's done so well for us this season. That, we believe, will give us the small edge that we need to maintain what we've been doing and Roberto [De Zerbi] understands the model.
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"He understands how we work, he supports the way we work and he's been one of the biggest advocates of bringing those young players into the first-team squad from the under-21s.
"Jeremy Sarmiento, Julio Enciso, just recently Facundo Buonanotte, just 18 and he's got himself into the Argentina senior squad for the first time.
"Roberto trusts their ability, trusts their confidence out on the pitch to do the job that perhaps more experienced players are doing in other clubs."
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Yves Bissouma and Leandro Trossard are two stars who have left in recent months, but Caicedo and Kaoru Mitoma have most certainly filled their voids.
Future planning is the key to Brighton's success as Barber added: "We try and get the players in the door before we need them because if you go out and sell a player for a lot of money and then go looking for his replacement, everything becomes that much more difficult.
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"We try and think ahead, we try and plan ahead as best we can, not just on the field but off the field. We're not perfect, we still make mistakes with our recruitment, we don't get everything right.
"Sometimes players just don't work out for whatever reason, it doesn't necessarily mean they're a bad player at all. Sometimes they just don't work out for family reasons, environment reasons, they don't get on with the coach, the teammates.
"We've had a couple of incidents of that so it's not perfect, but each year that goes by we feel the process is refined a little bit more. We get a little bit closer to the targets we really want and we fish in different ponds because we have to and we like to.
"We're very astute at finding young players that we feel are very coachable, that we can make better and that gives us an edge in terms not only when they're playing for us, but also when it comes potentially to selling them in the future.
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"That enables us to generate the profits that then allow us to do the same process again and keep doing it."