Exhibit 10.5

November 26, 2025

FACT II Acquisition Corp.

14 Wall Street, 20th Floor

New York, NY 10005

Attn: Adam Gishen, Chief Executive Officer

Ladies and Gentlemen:

This is to confirm our agreement whereby FACT II Acquisition Corp., a Cayman Islands exempted company (“Company”), has requested FACT II Acquisition Parent LLC, a Cayman Islands limited liability company (the “Advisor”) to serve as the Company’s advisor in connection with the Company effecting a merger, capital stock exchange, asset acquisition, stock purchase, reorganization or similar business combination or other strategic transactions (in each case, a “Strategic Transaction”).

| 1. | Services and Fees.

| (a) | The Advisor will, from time to time, upon the Company’s request and in consultation with the Company:

| (i) | assist the Company in preparing presentations for each potential Strategic Transaction;

| (ii) | introduce the Company to potential investors in connection with each potential Strategic Transaction;

| (iii) | assist the Company in arranging meetings with target company stockholders to discuss each potential Strategic Transaction, including any target company’s attributes and providing regular market feedback, including written status reports, from these meetings and participate in direct interaction with stockholders, in all cases to the extent legally permissible; and

| (iv) | assist the Company with the preparation of any press releases and filings related to each potential Strategic Transaction (the activities described in the foregoing clauses (i)-(iv), the “Services”).

| (b) | As compensation for the Services, Advisor will be entitled to a payment from the Company of up to $240,000 (which, in the sole discretion of the Company, may be payable in up to twelve monthly installments) following the date of the Company’s initial public offering (the “Fee”).

| (c) | The Fee shall be payable in cash and is due and payable in arrears at the Advisor’s request.

| 2. | Company Cooperation; Information.

| (a) | The Company will provide full cooperation to the Advisor as may be necessary for the efficient performance by the Advisor of its obligations hereunder, including, but not limited to, providing to the Advisor and its counsel, on a timely basis, all documents and information regarding the Company that the Advisor may reasonably request or that are otherwise relevant to the Advisor’s performance of its obligations hereunder (collectively, the “Information”); making the Company’s management, auditors, consultants and advisors available to the Advisor. The Company will promptly notify the Advisor of any change in facts or circumstances or new developments affecting the Company or that might reasonably be considered material to the Advisor’s engagement hereunder.

| (b) | The Advisor agrees to keep strictly confidential all information conveyed by the Company or the Company’s Representatives (as defined below) to the Advisor in connection with this Agreement including, for the avoidance of doubt, the identities of any potential parties to any such Strategic Transaction, in whatever form, whether written, electronic or oral, and to execute a non-disclosure agreement in customary form reasonably acceptable to the Advisor if requested to do so by the Company.

| 3. | Representations, Warranties, and Covenants.

| (a) | The Company represents, warrants and covenants to the Advisor that all Information it makes available to the Advisor by or on behalf of the Company in connection with the performance of its obligations hereunder will not contain any untrue statement of a material fact or omit to state a material fact necessary in order to make statements made, in light of the circumstances under which they were made, not misleading as of the date thereof. The Company acknowledges and agrees that the Advisor will use and rely on the accuracy and completeness of the Information supplied to the Advisor without having the obligation to independently verify the same.

| (b) | The Advisor represents, warrants and covenants to the Company that it is not prohibited from entering into this Agreement by any other contract, agreement, law or order.

| 4. | Trust Account.

The Advisor agrees (i) that it does not have any right, title, interest or claim of any kind in or to any monies in the Company’s trust account established in connection with its initial public offering (“Trust Account”) solely with respect to this Agreement (each, a “Claim”); (ii) to waive any Claim it may have against the Trust account in the future as a result of, or arising out of, any services provided to the Company hereunder; and (iii) to not seek recourse against the Trust Account with respect to the Fee.

| 5. | Use of Name and Reports.

Without the Advisor’s prior written consent, neither the Company nor any of its affiliates (nor any director, officer, manager, partner, member, employee, or agent thereof) shall quote or refer to (i) the Advisor’s name or (ii) any advice rendered by the Advisor to the Company or any communication from the Advisor in connection with performance of the Services, except as required by applicable federal or state law, regulation or securities exchange rule.

| 6. | Status as Independent Contractor.

The Advisor shall perform the Services as an independent contractor and not as an employee of the Company or affiliate thereof. It is expressly understood and agreed to by the parties that the Advisor shall have no authority to act for, represent or bind the Company or any affiliate thereof in any manner, except as may be expressly agreed to by the Company in writing. In rendering the Services, the Advisor will be acting solely pursuant to a contractual relationship on an arm’s-length basis.

| 7. | Entire Agreement.

This Agreement constitutes the entire understanding between the Company and Advisor with respect to the subject matter hereof and supersedes all prior agreements and understandings, oral or written, with respect thereto. This Agreement may not be modified or terminated orally or in any manner other than by an agreement in writing signed by the Company and the Advisor.

| 8. | Notices.

Any notices required or permitted to be given hereunder shall be in writing and shall be deemed given when mailed by certified mail or private courier service, return receipt requested, addressed to each party at its respective addresses set forth above, or such other address as may be given by a party in a notice given pursuant to this Section.

| 9. | Successors and Assigns.

This Agreement may not be assigned by any party without the written consent of the other parties. This Agreement shall be binding upon and shall inure to the benefit of the parties hereto and, except where prohibited, to their successors and assigns.

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| 10. | Non-Exclusivity.

Nothing herein shall be deemed to restrict or prohibit the engagement by the Company of other consultants providing the same or similar services or the payment by the Company of fees to such other consultants. The Company’s engagement of any other consultant(s) shall not affect the Advisor’s right to receive the Fee and reimbursement of expenses pursuant to this Agreement.

| 11. | Applicable Law; Venue.

This Agreement shall be construed and enforced in accordance with the laws of the State of New York without giving effect to conflict of laws. In the event of any dispute under this Agreement, then and in such event, each party hereto agrees that the dispute shall be brought and enforced in the courts of the State of New York, County of New York under the accelerated adjudication procedures of the Commercial Division, or the United States District Court for the Southern District of New York, in each event at the discretion of the party initiating the dispute. Each party irrevocably submits to such jurisdiction, which jurisdiction shall be exclusive. Each party hereby waives any objection to such exclusive jurisdiction and that such courts represent an inconvenient forum. Any such process or summons to be served upon a party may be served by transmitting a copy thereof by registered or certified mail, postage prepaid, addressed to such party at the address set forth at the beginning of this Agreement. Such mailing shall be deemed personal service and shall be legal and binding upon the party being served in any action, proceeding or claim. The parties agree that the prevailing party(ies) in any such action shall be entitled to recover from the other party(ies) all of its reasonable attorneys’ fees and expenses relating to such action or proceeding and/or incurred in connection with the preparation therefor.

| 12. | Counterparts.

This Agreement may be executed in several original or facsimile counterparts, each one of which shall constitute an original, and together shall constitute but one instrument.

[Signature Page Follows]

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If the foregoing correctly sets forth the understanding between the Advisor and the Company with respect to the foregoing, please so indicate your agreement by signing in the place provided below, at which time this letter shall become a binding contract.

| FACT II ACQUISITION PARENT LLC

| By: | /s/ Richard Nespola, Jr.
| Name: | Richard Nespola, Jr.
| Title: | Manager

AGREED AND ACCEPTED BY:

FACT II ACQUISITION CORP.

By: | /s/ Min Lee |
Name: | Min Lee |
Title: | Chief Financial Officer |

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