Am I being double taxed, or do I not understand?

[Kevin Robert Dean]

I just got my unemployment check and the fine print said I
could ask them to withhold 10% for Federal and State taxes,
or else I'll wind up paying more taxes come next years tax
time.

Am I being double taxed here?  Didn't I already pay for my
check with the unemployment insurance I paid into while I
was working, or am I just confused about how it works?
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