In baseball parlance, a bullpen is where relief pitchers warm up before going into the game. A team's relief pitchers are also referred to as the bullpen.
Bullpen Capital is named after the baseball term to indicate its place where startups can "pitch" themselves to the Bullpen VCs, after the seed round, and with enough practice and funds to go into later rounds. As I had written before when Bullpen first launched, you can also think of Bullpen as a place where startups "warm" up before they get back out to the field fully refreshed.
In this video, the SF Giants owner and Atlanta Braves pitcher explain how Bullpen Capital got its name. Given the San Francisco Giants' recent World Series win, it's timely.