oversee, official, govt office, rule, administer
監 centers on watching over something with authority: overseeing, administering, or ruling from an official position. The idea of a government office follows naturally from that supervisory role.
監 originally depicted a person bending over a vessel of water to see their reflection, combining 臣 (eye) and 皿 (dish). This gave the idea of looking closely, which later extended to overseeing or inspecting.
The top part looks like an eye (臣) peering down into a dish (皿) to inspect its contents. Picture an official bending over a bowl to oversee everything inside.
For カン, imagine a can of official documents that an overseer inspects: can -> カン, and the official looks down into the can like looking into a dish.
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supervision; control; superintendence; direction
monitoring; watching; observation; surveillance; guarding; supervision; lookout
inspection; audit; judgement; judgment
confinement; imprisonment; incarceration
(editorial) supervision; general editorship
prison
imprisonment
inspection; inspector
special Nara-period administrative division for areas containing a detached palace (Yoshino and Izumi)
inspector general; commissioner
superintendence; supervision; administration
Superintendent General of the Metropolitan Police
assistant director (in taking professional movies)
auditor; controller; comptroller
custody and care
supervising a translation
dormitory dean
(prison) cell; ward
online real-time surveillance
system auditing