Yes. The organelle replicates with the host cell. As time goes on more and more of the bacterium’s DNA involved in separate survival outside the host just goes

Yes. The organelle replicates with the host cell. As time goes on more and more of the bacterium’s DNA involved in separate survival outside the host just goes away as it further specializes in its task and leaves the rest to the host cell. Chloroplasts and mitochondria had the same kind of beginning.