Why does this case lead to a contradiction?

Question

From Mac Lane's Category theory:

Concerning the last line of the proof, I see why having card$(J) \gt$ card arr$(C)$ leads to a contradiction because it shows a category has more arrows that itself which is a contradiction, but why does the case where card$(J) \le$ card arr$(C)$ lead to a contradiction as well?

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Answer 1

The assumption that $C$ is not a preorder must lead here to a contradiction. This to justify the conclusion that $C$ is a preorder.

This can be done as sketched in the proof working with a small set $J$ that has a cardinality that exceeds the cardinality of the arrow set of $C$.