If, as we suppose, you have a cat, you will know that it is much more than just a domestic animal. Your cat is a member of the family, and just because he can't talk doesn't mean that you can't establish a very strong bond with him; which makes things more difficult when it comes time to say goodbye.
Remember that the sensation of pain is perfectly normal and that it is true that time heals everything.
For some people, it is really important to be able to say goodbye to their cat. When the vet comforts the cat to sleep, the vet will always allow you to stay with your cat. Even if you could not stay in the room at that time, you will have time to accompany your cat to say the last goodbye to him later.
Grief to smile
Following the death of a cat, you may feel a variety of emotions: depression, disbelief, grief, anger, guilty, anxiety, and finally acceptance. Don't repress any of these feelings; they are perfectly natural. Grief, like all human emotional processes, plays a very important role. It helps you accept death. Over time, memories that initially made you feel guilty or angry will begin to evoke it in a positive way, and you will find yourself smiling, not crying.
Help and support
Some people prefer to suffer their grief alone, but for others, it is important to have the support of family and friends. It may be embarrassing to show that you feel sorry "for a simple cat," but there is no reason for it. Still, it may be less difficult to discuss this with strangers, and in that case, your vet will likely be able to connect you with local support groups.
If you cannot share your pain with others, write your feelings and thoughts in a journal, you will see that you will feel better.