It's not uncommon for parents to want their former partner to approve of their babysitter. After all, you both likely have a vested interest in your child's well-being and want to make sure they're being cared for by someone you can trust. However, there are a few things to keep in mind if you find yourself in this situation.
For one, it's important to remember that you are the custodial parent and therefore have the final say in who watches your child. If your ex isn't on board with your choice of babysitter, you may need to have a conversation about why you feel they would be a good fit. It's also possible that your ex is simply uncomfortable with the idea of someone else caring for your child and may need some time to adjust.
Ultimately, it's up to you whether or not you want to involve your ex in the decision-making process when it comes to choosing a babysitter. However, it's important to communicate openly and honestly with them about your plans so that everyone is on the same page.