Cingulate Inc. Kansas City, Kansas 66205 Attention: Shane J. Schaffer, Chief Executive Officer Dear Mr. Schaffer:

Exhibit 10.5

May 2, 2023

Cingulate Inc.

0000 X. 00^xx Xxxxx

Kansas City, Kansas 66205

Attention: Xxxxx X. Xxxxxxxx, Chief Executive Officer

Dear Xx. Xxxxxxxx:

Reference is made to the At The Market Offering Agreement, dated as of January 3, 2023 (the “ATM Agreement”), between Cingulate Inc. (the “Company”) and X.X. Xxxxxxxxxx & Co., LLC (the “Manager”). This letter (the “Amendment”) constitutes an agreement between the Company and the Manager to amend the ATM Agreement as set forth herein. Defined terms that are used but not defined herein shall have the meanings ascribed to such terms in the ATM Agreement.

1. The defined term “Agreement” in the ATM Agreement is amended to mean the ATM Agreement as amended by this Amendment.

2. Section 4(h) of the ATM Agreement is hereby amended and restated as follows:

“Subsequent Equity Issuances. The Company shall not deliver any Sales Notice hereunder (and any Sales Notice previously delivered shall not apply during such three Business Days) for at least three (3) Business Days prior to any date on which the Company or any Subsidiary offers, sells, issues, contracts to sell, contracts to issue or otherwise disposes of, directly or indirectly, any other shares of Common Stock or any Common Stock Equivalents (other than the Shares), subject to Manager’s right to waive this obligation, provided that, without compliance with the foregoing obligation, the Company may issue and sell Common Stock pursuant to any employee equity plan, stock ownership plan or dividend reinvestment plan of the Company in effect at the time of the respective Sales Notice and the Company may issue Common Stock issuable upon the conversion or exercise of Common Stock Equivalents outstanding at the time of the respective Sales Notice, and provided, further, that the foregoing obligation shall not apply in connection with the Company’s sale of shares of Common Stock under an equity line of credit pursuant to a definitive agreement entered into on April 24, 2023 with Lincoln Park Capital, LLC (the “Equity Line”), provided that, on any single given Trading Day, the Company shall only deliver either (i) a Sales Notice for the sale of Shares hereunder or (ii) any notice for the sale of shares of Common Stock under the Equity Line.”

3. The Company and the Manager hereby agree that the date of this Amendment shall be a Representation Date under the ATM Agreement and the deliverables under Section 4(k), 4(l) and 4(m) of the ATM Agreement shall be required on the date of this Amendment.

4. The Agreement shall continue in full force and effect after the execution of this Amendment and shall not be in any way changed, modified or superseded by the terms set forth herein.

5. This Amendment may be executed in two or more counterparts and by facsimile or “.pdf” signature or otherwise, and each of such counterparts shall be deemed an original and all of such counterparts together shall constitute one and the same agreement.

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In acknowledgment that the foregoing correctly sets forth the understanding reached by the Company and the Manager, please sign in the space provided below, whereupon this Amendment shall constitute a binding amendment to the ATM Agreement as of the date indicated above.

	Very truly yours,
	X.X. XXXXXXXXXX & CO., LLC
	By:	/s/ Xxxx X. Xxxxxxx
	Name:	Xxxx X. Xxxxxxx
	Title:	Chief Executive Officer

Accepted and Agreed:

CINGULATE INC.

By:	/s/ Xxxxx Xxxxxxxx
Name:	Xxxxx Xxxxxxxx
Title:	Chief Executive Officer

[signature page to CING Amendment to atm agreement]

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