No, just install it on the iPhone the same way you did for the iPad, through the Apple app store. You can also access Lightroom on the web and see all your photos at: Lightroom.adobe.com You will have to log in with you Adobe account info.
@Chiara255403989l2q Ok, I have the same plan. I have LrC, PS and LR installed on 4 computers (only using 2 two at a time 🙂 ) and Lightroom also installed on iPhone, iPad, MBP and PC. You should have no restrictions on your LR installations as long as you are logging in properly with the Adobe ID used on the account. I get flawless if sometimes slow synching of all the photos in these apps and devices.