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Ian Price (Oct 25 2024 at 17:25):Chasing down the definition of Weakly LCCC from this paper https://arxiv.org/abs/1404.6997 leds to a remark 3.2 from https://www.sciencedirect.com/science/article/pii/S0022404999001929 and I'm blanking on seeing how the definition 3.1 and remark 3.2 are the same.
definition 3.1. says a category is weakly lccc if it has weak limits and every slice has weak simple products (defined on p2 of the second paper)
remark 3.2 says that if C has strong finite limits, then this condition can be expressed in terms of "weak representation"; for maps p: I -> J b : B -> I there is a map w: W -> J and a natural surjection C/J(-, w) ->> C/I(p*(-), b).
If anyone can explain why these are equivalent, I'd appreciate it