Hi,
The extension can be installed to multiple FI's the best way to identify the FI is to add the wedge_config field if it's a client-side extension or vendor_config if it is an SSO extension. the wedge configs /vendor configs will be saved in DataBase .so during deployment you can update the extension-related key and perform business logic over it.
Thanks,
Swarup
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Swarup's suggestion is a good one and easy enough to support if you need a specific value for each FI. But, if you just need a unique idetifier then we would suggest using the FI's ABA. The FI's ABA is available in self.hq_credentials.aba