3 USC 15 - Sec. 15. Counting electoral votes in Congress
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Upon such reading of any such certificate or paper, the President of the Senate shall call for objections, if any. Every objection shall be made in writing, and shall state clearly and concisely, and without argument, the ground thereof, and shall be signed by at least one Senator and one Member of the House of Representatives before the same shall be received.
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In the example you referenced I believe it was the Congresional Black Caucus who made the objection in the House and Senator Barbara Boxed joined the objection from the Senate.
What is the procedure after the objection is signed and received (by the President of the Senate)? Is it read to the joint session? Is it thrown open for discussion and debate?