Can I see my pet on my home screen?

On iPhone and iPad, yes. Long-press your home screen, tap the + button, search for WitnessWork, and choose Pet Companion. Small shows your pet; medium adds its name, hearts, your streak, and what its wall sign shows; large shows its whole room with its name and hearts. The widget moves through the day on its own, and opening WitnessWork refreshes it. Until you adopt a pet, it asks you to choose one in the app. Android doesn’t have WitnessWork home screen widgets.