No.
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The landlord’s responsibility is to provide correct information about the property and, when necessary, a properly prepared tenancy-related document that can be used for registration.
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HomeInZagreb’s responsibility is to act as an agency, help organise your accommodation and inform you in general about the need to register.
But:
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neither the landlord nor HomeInZagreb can be held responsible for your failure to register,
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nor for any issues with your visa, residence permit, or legal status in Croatia or the EU.
These matters are governed exclusively by Croatian and EU laws and are under the authority of public institutions, not private companies or landlords.