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@ayegill @nilesjohnson @Zanzi @hallasurvivor
Ah, yes. This was confusing me for a while when I first saw it.

On a shallow level I understand that coend formulation of weighted colimits uses copowers (which are covariant in both variables) and that end formulation uses powers (which are contravariant in one variable).
This then means that contravariance (as necessitated by the coend) has to go somewhere, and it goes in the weight.

But outside of the coend formulaton I don't intuitively understand why colimits would need contravariant weights.