Under California law either or both companies could be "your employer" depending on nature of a claim you might be asserting. For taxes and withholding, it's normally the employment agency, not the client, who is paying employment taxes and benefit premiums, as said above. Clearly, both won't be paying, so the contractual relationship between the two will specify who the payrolled employee belongs to for tax and premium purposes. Attorneys will often name both companies in a claim to prevent one or the other from slipping off the hook if no company admits to being the accountible employer.
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It depends what you are really asking. The employer could be one or the other entity for some purposes and a joint employer for other purposes.
If you are asking who collects the employee's withholding and deposits the withholding to the IRS, then it is the entity that issues the paychecks. If this isn't what you want to know, then we need more detail to answer the question.
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Employment agencies customarily hire and pay employees for temp positions. The employee is on their payroll because the agency normally takes a fee from the employer/company.
Whoever issues the paycheck - is the employer, and it is unlikely that the both are doing this.
This is my opinion and should not be construed as legal advise for your specific case as there are many more facts which you have not provided.