Because he became a member of the Kobayakawa clan before his genpuku (celebrate one's coming of age), his imina (personal name) doesn't use the letter 'moto' (元), which is a tsuji (distinctive character used in the names of all people belonging to a single clan or lineage) of the Mori clan.
元服する前に小早川氏に入ったため、彼の諱には毛利氏の通字「元」が含まれていない。