I'm confused on the editorial for 1666J - Job Lookup. View the editorial here

In the editorial for 1666J - Job Lookup, when transitioning from

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and

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to

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$$$c_{u, v}$$$ seems to be only added once (the number of messages going from $$$u$$$ to $$$v$$$, for all $$$u,v$$$ such that

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and

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) even though the total cost should be

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. What is going on here?

Thank you for the help!