sponsor, hold (a meeting), give (a dinner)
催 centers on causing an event to happen by bringing people together: sponsoring, holding a meeting, or giving a dinner. The common thread is organizing and prompting a gathering.
催 combines 亻 (person) with 崔, which likely contributes the sound and carries a sense of urging or pressing. The exact historical development of 崔 is uncertain, but the character is used for prompting or organizing events.
A person 亻 standing beside 崔, which looks like a mountain 山 over a bird 隹, as if urging the bird to fly. Picture someone organizing a mountain-top meeting where birds are prompted to gather.
For サイ, imagine a sponsor sighing with relief after a successful dinner: sigh -> サイ, and the person 亻 has just held the event.
holding (a conference, exhibition, etc.); opening; hosting (e.g. the Olympics)
sponsorship (i.e. conducting under one's auspices); promotion; organizing; organising; hosting; staging
hypnosis
event; festivities; function; social gathering; auspices; opening; holding (a meeting)
to hold (an event); to give (a dinner, party, etc.)
pressing; urging; demanding; demand
joint sponsorship (of an event); cosponsorship; joint hosting
tear-inducing
special event
(program of) entertainments; special events; exhibitions; amusements; attractions
carrying out a trip (according to plans)
arousing sexual desire
notification
combining a pair of events; joint
sponsoring nation
sponsor; promoter; impresario; organizer; organiser
to hold a meeting
galactogogue; galactagogue
event hall
dun; letter requesting money, etc.