No, I do not let my cat outside to play and explore. I keep her indoors for her safety and to protect her from potential predators. I do, however, provide her with plenty of toys and activities to keep her entertained and stimulated.
It's important to let your cat explore outside in their own time. We wouldn't recommend using a harness, and you shouldn't pick your cat up and carry them outside. Both of these situations could potentially cause your cat stress and may result in them associating this negative feeling with going or being outside.
It's good to let your cat play outside but also ensure that it's not risky such the presence of stray cats and dogs, and harmful people who can harm it.
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