Warning after two years:
Please take this proof with a grain of salt. This proof is long and has lots of details, so expect lots of typos. Two years have passed, and things didn’t go well. I still believe the idea is correct (but inefficient) though.
I had difficulties with this problem, and found someone already posted a question on MSE with no categorical answer present. (UPDATE: Now there is a good one thanks to Yizhen Chen.) After a week of frustration, I came up with this fully categorical approach:
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