Exhibit 10.4


 


July
26, 2021


 


Alpha
Healthcare Acquisition Corp. III


1177
Avenue of the Americas, 5th Floor


New
York, New York 10036


 


	Re:	Initial
    Public Offering


 


Ladies
and Gentlemen:


 


This
letter (this “Letter Agreement”) is being delivered to you in accordance with the Underwriting
Agreement (the “Underwriting Agreement”) entered into by and among Alpha Healthcare Acquisition Corp.
III, a Delaware corporation (the “Company”), and BofA Securities, Inc. and PJT Partners LP, as representatives
(the “Representatives”) of the several underwriters (each, an “Underwriter” and collectively,
the “Underwriters”), relating to an underwritten initial public offering (the “Public Offering”),
of 15,000,000 of the Company’s units (including up to 2,250,000 units that may be purchased to cover over-allotments, if any) (the
“Units”), each comprised of one share of the Company’s Class A common stock, par value $0.0001 per share
(the “Common Stock”), and one-fourth of one redeemable warrant. Each whole warrant (each,
a “Warrant”) entitles the holder thereof to purchase one share of Common Stock at a price of $11.50 per share,
subject to adjustment. The Units will be sold in the Public Offering pursuant to a registration statement on Form S-1 (File No. 333-253876)
and prospectus (the “Prospectus”) filed by the Company with the U.S. Securities and Exchange Commission (the
“Commission”) and the Company has applied to have the Units listed on The Nasdaq Capital Market. Certain capitalized
terms used herein are defined in paragraph 11 hereof.


 


In
order to induce the Company and the Underwriters to enter into the Underwriting Agreement and to proceed with the Public Offering and
for other good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, each of AHAC Sponsor III LLC
(the “Sponsor”) and the undersigned individuals, each of whom is a member of the Company’s board of directors
and/or management team or an advisor of the Company (each, an “Insider” and collectively, the “Insiders”),
hereby agrees with the Company as follows:


 


1.
The Sponsor and each Insider agrees that if the Company seeks stockholder approval of a proposed Business Combination, then in connection
with such proposed Business Combination, it, he or she shall (i) vote any shares of Capital Stock owned by it, him or her in favor of
any proposed Business Combination and (ii) not redeem any shares of Common Stock owned by it, him or her in connection with such stockholder
approval. If the Company engages in a tender offer in connection with any proposed Business Combination, the Sponsor and each Insider
agrees that it, he or she will not seek to sell or tender any shares of Capital Stock owned by it, him or her to the Company in connection
with such tender offer.


 


2.
The Sponsor and each Insider hereby agrees that in the event that the Company fails to consummate a Business Combination 24 months from
the closing of the Public Offering, or such later period approved by the Company’s stockholders in accordance with the Company’s
amended and restated certificate of incorporation (as it may be amended from time to time, the “Charter”),
the Sponsor and each Insider shall take all reasonable steps to cause the Company to (i) cease all operations except for the purpose
of winding up, (ii) as promptly as reasonably possible but not more than 10 business days thereafter, subject to lawfully available funds
therefor, redeem 100% of the shares of Common Stock sold as part of the Units in the Public Offering (the “Offering Shares”),
at a per-share price, payable in cash, equal to the aggregate amount then on deposit in the Trust Account (as defined below), including
interest earned on the funds held in the Trust Account and not previously released to the Company to pay its taxes (less up to $100,000
of interest to pay dissolution expenses), divided by the number of then outstanding Offering Shares, which redemption will completely
extinguish the rights of all holders of Offering Shares as stockholders (including the right to receive further liquidating distributions,
if any), subject to applicable law, and (iii) as promptly as reasonably possible following such redemption, subject to the approval of
the Company’s remaining stockholders and the Company’s board of directors, dissolve and liquidate, subject in the case of
clauses (ii) and (iii) to the Company’s obligations under Delaware law to provide for claims of creditors and the requirements
of other applicable law. The Sponsor and each Insider agrees not to propose any amendment to the Charter to modify (i) the substance
or timing of the ability of holders of Offering Shares to seek redemption in connection with the Company’s initial Business Combination
or amendments to the Charter prior thereto or (ii) (A) the Company’s obligation to redeem 100% of the Offering Shares if the Company
does not complete its initial Business Combination within such time set forth in the Charter or (B) with respect to any other provision
relating to stockholders' rights or pre-initial Business Combination activity, unless the Company provides the holders of the Offering
Shares with the opportunity to redeem their shares of Common Stock upon approval of any such amendment at a per-share price, payable
in cash, equal to the aggregate amount then on deposit in the Trust Account, including interest earned on the funds held in the Trust
Account and not previously released to the Company to pay its taxes, divided by the number of then outstanding Offering Shares.


 



     


     


    

 


        The
Sponsor and each Insider acknowledges that it, he or she has no right, title, interest or claim of any kind in or to any monies held
in the Trust Account or any other asset of the Company as a result of any liquidation of the Company with respect to the Founder Shares
or Private Placement Shares held by it, him or her. The Sponsor and each Insider hereby further waives, with respect to any shares of
Common Stock held by it, him or her, if any, whether acquired now or hereafter, any redemption rights it, he or she may have in connection
with the consummation of a Business Combination or amendments to the Charter prior thereto, including, without limitation, any such rights
available in the context of a stockholder vote to approve such Business Combination or a stockholder vote to approve an amendment to
the Charter to modify (i) (A) the substance or timing of the Company’s obligation to redeem 100% of the Offering Shares if the
Company has not consummated a Business Combination within the time period set forth in the Charter or (B) any other provisions relating
to stockholders’ rights or pre-initial Business Combination activity or (ii) in the context of a tender offer made by the Company
to purchase shares of Common Stock (although the Sponsor, the Insiders and their respective affiliates shall be entitled to redemption
and liquidation rights with respect to any Offering Shares it or they hold if the Company fails to consummate a Business Combination
within the time period set forth in the Charter).


 


3.
During the period commencing on the effective date of the Underwriting Agreement and ending 180 days after such date, the Sponsor and
each Insider shall not, without the prior written consent of the Representatives, (i) sell, offer to sell, contract or agree to sell,
hypothecate, pledge, grant any option to purchase or otherwise dispose of or agree to dispose of, directly or indirectly, or establish
or increase a put equivalent position or liquidate or decrease a call equivalent position within the meaning of Section 16 of the Securities
Exchange Act of 1934, as amended (the “Exchange Act”), and the rules and regulations of the Commission promulgated
thereunder, with respect to any Units, shares of Capital Stock, Warrants or any securities convertible into, or exercisable, or exchangeable
for, shares of Capital Stock owned by it, him or her, (ii) enter into any swap or other arrangement that transfers to another, in whole
or in part, any of the economic consequences of ownership of any Units, shares of Capital Stock, Warrants or any securities convertible
into, or exercisable, or exchangeable for, shares of Capital Stock owned by it, him or her, whether any such transaction is to be settled
by delivery of such securities, in cash or otherwise, or (iii) publicly announce any intention to effect any transaction specified in
clause (i) or (ii). Each of the Insiders and the Sponsor acknowledges and agrees that, prior to the effective date of any release or
waiver, of the restrictions set forth in this paragraph 3 or paragraph 7 below, the Company shall announce the impending release or waiver
by press release through a major news service at least two business days before the effective date of the release or waiver. Any release
or waiver granted shall only be effective two business days after the publication date of such press release. The provisions of this
paragraph will not apply if the release or waiver is effected solely to permit a transfer not for consideration and the transferee has
agreed in writing to be bound by the same terms described in this Letter Agreement to the extent and for the duration that such terms
remain in effect at the time of the transfer.


 


4.
In the event of the liquidation of the Trust Account upon the failure of the Company to consummate its initial Business Combination within
the time period set forth in the Charter, the Sponsor (the “Indemnitor”) agrees to indemnify and hold harmless
the Company against any and all loss, liability, claim, damage and expense whatsoever (including, but not limited to, any and all legal
or other expenses reasonably incurred in investigating, preparing or defending against any litigation, whether pending or threatened)
to which the Company may become subject as a result of any claim by (i) any third party for services rendered or products sold to the
Company or (ii) any prospective target business with which the Company has entered into a written letter of intent, confidentiality or
other similar agreement or Business Combination agreement (a “Target”); provided, however, that such
indemnification of the Company by the Indemnitor shall (x) apply only to the extent necessary to ensure that such claims by a third party
or a Target do not reduce the amount of funds in the Trust Account to below the lesser of (i) $10.00 per Offering Share and (ii) the
actual amount per Offering Share held in the Trust Account as of the date of the liquidation of the Trust Account, if less than $10.00
per Offering Share is then held in the Trust Account due to reductions in the value of the trust assets, less interest earned on the
Trust Account which may be withdrawn to pay taxes, (y) not apply to any claims by a third party or a Target which executed a waiver of
any and all rights to the monies held in the Trust Account (whether or not such waiver is enforceable) and (z) not apply to any claims
under the Company’s indemnity of the Underwriters against certain liabilities, including liabilities under the Securities Act of
1933, as amended. The Indemnitor shall have the right to defend against any such claim with counsel of its choice reasonably satisfactory
to the Company if, within 15 days following written receipt of notice of the claim to the Indemnitor, the Indemnitor notifies the Company
in writing that it shall undertake such defense.


  



    2


     


    

 


5.
To the extent that the Underwriters do not exercise their over-allotment option to purchase up to an additional 2,250,000 Units in full
within 45 days from the date of the Prospectus (and as further described in the Prospectus), the Sponsor agrees to forfeit, at no cost,
a number of Founder Shares in the aggregate equal to 562,500 multiplied by a fraction, (i) the numerator of which is 2,250,000 minus
the number of Units purchased by the Underwriters upon the exercise of their over-allotment option, and (ii) the denominator of which
is 2,250,000. The forfeiture will be adjusted to the extent that the over-allotment option is not exercised in full by the Underwriters
so that the Sponsor will be required to forfeit only that number of Founder Shares as is necessary so that the Initial Stockholders will
own an aggregate of 20.0% of the Company’s issued and outstanding shares of Capital Stock after the Public Offering (not including
the Private Placement Shares).


 


6.
The Sponsor and each Insider hereby agrees and acknowledges that: (i) the Underwriters and the Company would be irreparably injured in
the event of a breach by such Sponsor or an Insider of its, his or her obligations under paragraphs 1, 2, 3, 4, 5, 7(a), 7(b), and 9,
as applicable, of this Letter Agreement (ii) monetary damages may not be an adequate remedy for such breach and (iii) the non-breaching
party shall be entitled to injunctive relief, in addition to any other remedy that such party may have in law or in equity, in the event
of such breach.


 


7.
(a) The Sponsor and each Insider agrees that it, he or she shall not Transfer any Founder Shares (or shares of Common Stock issuable
upon conversion thereof) until the earlier to occur of (A) one year after the completion of the Company’s initial Business Combination
and (B) subsequent to the Company’s initial Business Combination, (x) if the reported last sale price of the Common Stock equals
or exceeds $12.00 per share (as adjusted for stock splits, stock dividends, reorganizations, recapitalizations and the like) for any
20 trading days within any 30-trading day period commencing at least 150 days after the Company’s initial Business Combination,
or (y) the date on which the Company completes a liquidation, merger, capital stock exchange, reorganization or other similar transaction
that results in all of the Company’s stockholders having the right to exchange their shares of Common Stock for cash, securities
or other property (the “Founder Shares Lock-up Period”).


 


(b)
The Sponsor and each Insider agrees that it, he or she shall not Transfer any Private Placement Units, the Private Placement Shares,
the Private Placement Warrants or shares of Common Stock issued or issuable upon the exercise of the Private Placement Warrants, until
30 days after the completion of the initial Business Combination (the “Private Placement Units Lock-up Period”,
together with the Founder Shares Lock-up Period, the “Lock-up Periods”).


 


(c)
Notwithstanding the provisions set forth in paragraphs 7(a) and 7(b), Transfers of the Founder Shares, Private Placement Units, Private
Placement Shares, Private Placement Warrants and shares of Common Stock issued or issuable upon the exercise or conversion of the Private
Placement Warrants or the Founder Shares that are held by the Sponsor, any Insider or any of their permitted transferees (that have complied
with this paragraph 7(c)), are permitted (i) to the Company’s officers or directors, any affiliate or family member of any of the
Company’s officers or directors or any members of the Sponsor or any affiliates of the Sponsor; (b) in the case of an individual,
by gift to a member of such individual’s immediate family or to a trust, the beneficiary of which is a member of such individual’s
immediate family, an affiliate of such individual or to a charitable organization; (c) in the case of an individual, by virtue of laws
of descent and distribution upon death of such individual; (d) in the case of an individual, pursuant to a qualified domestic relations
order; (e) by private sales or transfers made in connection with the consummation of an initial Business Combination at prices no greater
than the price at which the securities were originally purchased; (f) in the event of the Company’s liquidation prior to the completion
of an initial Business Combination; (g) by virtue of the laws of the State of Delaware or the Sponsor’s limited liability company
agreement upon dissolution of the Sponsor; or (h) in the event of the Company’s liquidation, merger, capital stock exchange,
reorganization or other similar transaction which results in all of the Company’s stockholders having the right to exchange their
shares of Common Stock for cash, securities or other property subsequent to the Company’s completion of an initial Business Combination;
provided, however, that in the case of clauses (a) through (e) or (g), these permitted transferees must enter into a written
agreement with the Company agreeing to be bound by the transfer restrictions herein and the other restrictions contained in this Agreement
(including provisions relating to voting, the Trust Account and liquidating distributions).


 



    3


     


    

 


8.
The Sponsor and each Insider represents and warrants that it, he or she has never been suspended or expelled from membership in any securities
or commodities exchange or association or had a securities or commodities license or registration denied, suspended or revoked. Each
Insider’s biographical information furnished to the Company (including any such information included in the Prospectus) is true
and accurate in all respects and does not omit any material information with respect to the Insider’s background. Each Insider’s
questionnaire furnished to the Company is true and accurate in all respects. Each Insider represents and warrants that: it, he or she
is not subject to or a respondent in any legal action for, any injunction, cease-and-desist order or order or stipulation to desist or
refrain from any act or practice relating to the offering of securities in any jurisdiction; it, he or she has never been convicted of,
or pleaded guilty to, any crime (i) involving fraud, (ii) relating to any financial transaction or handling of funds of another person,
or (iii) pertaining to any dealings in any securities and it, he or she is not currently a defendant in any such criminal proceeding.


 


9.
Except as disclosed in the Prospectus, neither the Sponsor nor any officer or director of the Company nor any affiliate of the Sponsor
nor any affiliate of any officer or director of the Company, shall receive from the Company any finder’s fee, reimbursement, consulting
fee, monies in respect of any repayment of a loan or other compensation prior to, or in connection with any services rendered in order
to effectuate, the consummation of the Company’s initial Business Combination (regardless of the type of transaction that it is),
, other than the following, none of which will be made from the proceeds held in the Trust Account prior to the completion of the initial
Business Combination: repayment of up to an aggregate of $300,000 in loans made to the Company by an affiliate of the Sponsor to cover
offering-related and organizational expenses; payment to an affiliate of the Sponsor of $10,000 per month, for up to 24 months, for office
space, utilities and secretarial and administrative support; reimbursement for any reasonable out-of-pocket expenses related to identifying,
investigating and completing an initial Business Combination; and repayment of non-interest bearing loans which may be made by the Sponsor
or an affiliate of the Sponsor or certain of the Company’s officers and directors to finance transaction costs in connection with
an intended initial Business Combination, the terms of which (other than as described above) have not been determined nor have any written
agreements been executed with respect thereto. Up to $1,500,000 of such loans may be convertible into units, at a price of $10.00 per
unit at the option of the lender, upon consummation of the initial Business Combination. The units would be identical to the Private
Placement Units.


 


10.
The Sponsor and each Insider has full right and power, without violating any agreement to which it is bound (including, without limitation,
any non-competition or non-solicitation agreement with any employer or former employer), to enter into this Letter Agreement and, as
applicable, to serve as an officer and/or director on the board of directors of the Company and hereby consents to being named in the
Prospectus as an officer and/or director of the Company.


 


11.
As used herein, (i) “Business Combination” shall mean a merger, capital stock exchange,
asset acquisition, stock purchase, reorganization or similar business combination, involving the Company and one or more businesses;
(ii) “Capital Stock” shall mean, collectively, the Common Stock and the Founder Shares;
(iii) “Founder Shares” shall mean (a) the 4,312,500 shares of the Company’s Class
B common stock, par value $0.0001 per share, initially issued to the Sponsor (up to 562,500 Shares of which are subject to complete or
partial forfeiture by the Sponsor if the over-allotment option is not exercised by the Underwriters) for an aggregate purchase price
of $25,000, or $0.0057971 per share, prior to the consummation of the Public Offering; (iv) “Initial Stockholders”
shall mean the Sponsor and any Insider that holds Founder Shares; (v) “Private Placement Shares”
shall mean the 455,000 shares of Common Stock comprising the Private Placement Units (or up to 500,000 shares of Common Stock if the
over-allotment option is exercised in full); (vi) “Private Placement Units”
shall mean the 455,000 units, or up to 500,000 units if the over-allotment option is exercised in full, each comprised of one share of
Common Stock and one-fourth of one warrant to purchase one share of Common Stock, that the Sponsor has agreed to purchase for an aggregate
purchase price of $4,550,000 (or up to $5,000,000 if the over-allotment option is exercised in full), or purchase price of $10.00 per
Private Placement Unit, in a private placement that shall occur simultaneously with the consummation of the Public Offering; (vii) “Private Placement Warrants”
shall mean the Warrants to purchase up to 113,750 shares of Common Stock (or up to 125,000 shares of Common Stock if the over-allotment
option is exercised in full) comprising the Private Placement Units; (viii) “Public Stockholders”
shall mean the holders of securities issued in the Public Offering; (ix) “Trust Account”
shall mean the trust fund into which a portion of the net proceeds of the Public Offering shall be deposited; and (x) “Transfer”
shall mean the (a) sale of, offer to sell, contract or agreement to sell, hypothecate, pledge, grant of any option to purchase or otherwise
dispose of or agreement to dispose of, directly or indirectly, or establishment or increase of a put equivalent position or liquidation
with respect to or decrease of a call equivalent position within the meaning of Section 16 of the Exchange Act, and the rules and regulations
of the Commission promulgated thereunder with respect to, any security, (b) entry into any swap or other arrangement that transfers to
another, in whole or in part, any of the economic consequences of ownership of any security, whether any such transaction is to be settled
by delivery of such securities, in cash or otherwise, or (c) public announcement of any intention to effect any transaction specified
in clause (a) or (b).


 



    4


     


    

 


12.
The Company will maintain an insurance policy or policies providing directors’ and officers’ liability insurance, and each
Director shall be covered by such policy or policies, in accordance with its or their terms, to the maximum extent of the coverage available
for any of the Company’s directors or officers.


 


13.
This Letter Agreement constitutes the entire agreement and understanding of the parties hereto in respect of the subject matter hereof
and supersedes all prior understandings, agreements, or representations by or among the parties hereto, written or oral, to the extent
they relate in any way to the subject matter hereof or the transactions contemplated hereby. This Letter Agreement may not be changed,
amended, modified or waived (other than to correct a typographical error) as to any particular provision, except by a written instrument
executed by all parties hereto.


 


14.
No party hereto may assign either this Letter Agreement or any of its rights, interests, or obligations hereunder without the prior written
consent of the other parties. Any purported assignment in violation of this paragraph shall be void and ineffectual and shall not operate
to transfer or assign any interest or title to the purported assignee. This Letter Agreement shall be binding on the Sponsor and each
Insider and their respective successors, heirs and assigns and permitted transferees.


 


15.
Nothing in this Letter Agreement shall be construed to confer upon, or give to, any person or corporation other than the parties hereto
any right, remedy or claim under or by reason of this Letter Agreement or of any covenant, condition, stipulation, promise or agreement
hereof. All covenants, conditions, stipulations, promises and agreements contained in this Letter Agreement shall be for the sole and
exclusive benefit of the parties hereto and their successors, heirs, personal representatives and assigns and permitted transferees.


 


16.
This Letter Agreement may be executed in any number of original or facsimile counterparts and each of such counterparts shall for all
purposes be deemed to be an original, and all such counterparts shall together constitute but one and the same instrument.


 


17.
This Letter Agreement shall be deemed severable, and the invalidity or unenforceability of any term or provision hereof shall not affect
the validity or enforceability of this Letter Agreement or of any other term or provision hereof. Furthermore, in lieu of any such invalid
or unenforceable term or provision, the parties hereto intend that there shall be added as a part of this Letter Agreement a provision
as similar in terms to such invalid or unenforceable provision as may be possible and be valid and enforceable.


 


18.
This Letter Agreement shall be governed by and construed and enforced in accordance with the laws of the State of New York, without giving
effect to conflicts of law principles that would result in the application of the substantive laws of another jurisdiction. The parties
hereto (i) all agree that any action, proceeding, claim or dispute arising out of, or relating in any way to, this Letter Agreement shall
be brought and enforced in the courts of New York City, in the State of New York, and irrevocably submit to such jurisdiction and venue,
which jurisdiction and venue shall be exclusive and (ii) waive any objection to such exclusive jurisdiction and venue or that such courts
represent an inconvenient forum.


 


19.
Any notice, consent or request to be given in connection with any of the terms or provisions of this Letter Agreement shall be in writing
and shall be sent by express mail or similar private courier service, by certified mail (return receipt requested), by hand delivery
or facsimile transmission.


 


20.
This Letter Agreement shall terminate on the earlier of (i) the expiration of the Lock-up Periods or (ii) the liquidation of the Company;
provided, however, that this Letter Agreement shall earlier terminate in the event that the Public Offering is not consummated and closed
by December 31, 2021; provided further that paragraph 4 of this Letter Agreement shall survive such liquidation.


 


21.
The Company, the Sponsor and each Insider hereby acknowledges and agrees that the Representatives on behalf of the Underwriters are each
a third party beneficiary of this Letter Agreement.


 


[Signature
Page Follows]


 



    5


     


    

  


	 	Sincerely,
	 	 
	 	AHAC
    SPONSOR III LLC
	 	 	 
	 	By:	 /s/
    Rajiv Shukla
	 	 	Name:
    Rajiv Shukla
	 	 	Title:   Managing
    Member


 


	 	By:	 /s/
    Rajiv Shukla
	 	 	Name:
    Rajiv Shukla


 


	 	By:	 /s/
    Patrick A. Sturgeon
	 	 	Name:
    Patrick A. Sturgeon


 


	 	By:	 /s/
    Darlene T. DeRemer
	 	 	Name:
    Darlene T. DeRemer


 


	 	By:	 /s/
    Eugene Podsiadlo 
	 	 	Name:
    Eugene Podsiadlo


 


	 	By:	 /s/
    William Woodward
	 	 	Name:
    William Woodward


   


	Acknowledged
    and Agreed:	 
	 	 
	ALPHA HEALTHCARE ACQUISITION CORP. III
	 	 	 
	By:	/s/
Rajiv Shukla	 
	 	Name: 	Rajiv Shukla	 
	 	Title:  	Chief Executive Officer	 


  


 


[Signature
Page to Letter Agreement]