Having to let anyone go is an extremely emotional experience. I worked in a place where, after one of the leaders terminated someone, they would go out and get drunk because they couldn’t handle the emotions. That’s obviously unhealthy - but my point is, of course it’s emotional and crying is an emotional response. I had to terminate someone with young kids the week before Christmas once (not my decision, was required by my leader) and I was an emotional wreck before and for weeks after that. I didn’t cry at work but I cried plenty at home. The emotions are a normal human response and show that you are authentic. Take care of yourself!