Exhibit
10.4


 


[●],
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., 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-          ) 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.


 



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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).


 






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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 $0.00579 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).


 






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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.


 






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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 third party beneficiaries of this Letter Agreement.


 


 


[Signature
Page Follows]


 






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



 

	 	By:	 
	 	 	Name: 	Rajiv Shukla



 

	 	By:	 
	 	 	Name: 	Patrick A. Sturgeon



 

	 	By:	 
	 	 	Name: 	Darlene T. DeRemer



 

	 	By:	 
	 	 	Name: 	Eugene Podsiadlo



 

	 	By:	 
	 	 	Name: 	William Woodward


  


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


 


 


[Signature
Page to Letter Agreement]


 



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