Pet Policy question - Grandfathered in pets

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Good morning! We're closing tomorrow on a 4-plex. The current owner and lease allows for tenants to have pets. My policy is no pets, and when the current tenants sign a new lease, it will be a "no-pets" lease. Also, when units turnover, the new tenants will not be allowed pets.

I realize that their current pets (1 tenant has a dog, another has 1 dog and 1 cat) will most likely need to be grandfathered in if I want to keep the tenants... as a pet owner myself, if it were me: I would find another place to live if all of a sudden my dog wasn't allowed. 

So I guess my question is this: When I bring in a new tenant to the empty unit and they sign a "no-pets" lease, do I need to explain that the other tenants have the same lease, but they were grandfathered in, so they can have pets but this new person can't, even though they have the same lease? How does this all work when transitioning an entire building to a no-pets policy?

Confusing, I know. Hope my question makes sense!

Thanks in advance!!