exam, school, printing, proof, correction
校 centers on systematic examination and correction, which naturally extends to the place where such checking happens: a school. The printing and proof senses are specific applications of comparing and correcting.
校 combines 木 (tree, wood) with 交 (exchange, mix), originally referring to wooden fetters or a framework for comparison. The modern meanings of school and examination developed from the idea of comparing and correcting texts.
A wooden 木 framework 交 where texts are compared and corrected: that's a school, where exams and proofreading happen.
For コウ, imagine a school corridor where students line up to hand in their corrected exams: corridor -> コウ, and the wooden 木 desks are where they check 交 their work.
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school
senior high school; high school
primary school; elementary school; grade school
senior high school student
junior high school; middle school; lower secondary school
principal; head teacher; headmaster; headmistress
attendance (at school); going to school
commissioned officer; military officer
school building; schoolhouse
preparatory school (for entrance examinations); cram school; cramming school
alma mater
within a school
main school; principal school
opening a school
school rules; school regulations
(temporary) closure of school
proofreading
the whole school
schoolyard; school playground; school grounds; campus
school song; college song; school anthem
the same school
school gate
revision; proofreading
branch school