Well spotted. The mistake in the ’proof’ is in confusing cones over and cones over . I think they are the same iff the right square is a pullback.
Edit: I made the same mistake at comma object. Both fixed now, I think.
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A --> B --> C
| | |
v v v
D --> E --> F
A ==== A ==== A
| | |
| |i |
| v |
A ---> B ---> A
i p
Added a proof of the pasting lemma to pullback, and the corresponding lemma to comma object (also added the construction by pullbacks and cotensors there).
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