継子 — 継 (inherit / succeed to) + 子 (child) — is the in-universe term for a Demon Slayer Corps junior whom a Hashira has formally taken on as a personal apprentice. The relationship is closer to a Japanese martial-arts 内弟子 (uchi-deshi, live-in disciple) than to a Western 'student': the Hashira is responsible for the tsuguko's training, performance, and survival, and the tsuguko is the publicly designated heir to the Hashira's seat. Most Hashira do not have a tsuguko; finding one with the right breath compatibility is rare, and a Hashira whose tsuguko dies before promotion often does not take another.

Kyojuro Rengoku does not have a tsuguko at the time of the Mugen Train arc — a fact that becomes structurally important in his death scene, where he addresses Tanjiro, Inosuke and Zenitsu as if they were his three tsuguko by spirit if not by paperwork. The English subtitle most commonly renders the word as 'successor' or 'apprentice,' both of which lose the kinship register of 子 ('child'). The Hashira-tsuguko bond is named in family vocabulary on purpose: it's meant to feel like a father-son relationship, the way martial lineages are often described in Japanese.