Being convicted in the Senate, yes. That would prohibit anyone from becoming President again. They could, however, still "run" for President. They just couldn't "be" President again.
Judgment in cases of impeachment shall not extend further than to removal from office, and disqualification to hold and enjoy any office of honor, trust or profit under the United States: but the party convicted shall nevertheless be liable and subject to indictment, trial, judgment and punishment, according to law.
The impeachment show had two objectives.
1st, was to hope/pressure the senate to take up the trial in the hopes of convicting him...thus preventing him from holding the office again.
2nd, if that didn't work...it was to place the label on him of being the only President in history to have been impeached twice.
Pelosi “This president is impeached for life, regardless of any gamesmanship on the part of Mitch McConnell,” the California Representative said, firming up Rep. Adam Schiff’s suggestion from a month ago that a failed Senate trial, by its very nature of existing, would not be a failure. “He will be impeached forever,” - Jan 2020.
Disqualifictaion is a separate punishment from removal.
Alcee Hastings was impeached and convicted, and removed from his Federal judgeship for bribery, but the punishment of disqualification was not imposed by Senate and Hastings sits in the House of Representatives to this very day, having been sent there repeatedly by the people of CD FL- 20.
In a sane world, future textbooks will cite the two impeachments of Donald Trump as evidence of extreme partisanship by the Democratic Party. Instead, no doubt, they will just cite them as showing how horrible he was.
I'm wondering what tack Biden will take in his inaugural address. Will it be meaningless platitudes and fuzzy statements about how much better things are about to become, or will it be nasty partisan rhetoric, saying that Trump tried to steal the election, incited insurrection, won the first time only because of Russian collusion, called neo-Nazis "very fine people," caused hundreds of thousands to die by mismanaging the corona virus, supported white supremacy and Islamophobia, etc., etc.? For good measure he could blast the Republicans for failing to vote to remove Trump a year ago.