Children may be afraid of animals, especially those they see with any regularity. But this fear is not something innate but learned. A child's fear of cats is usually the result of (1) a personal traumatic experience with a cat (shocked, scratched or bitten, for example), (2) a vicarious trauma (witnessing a traumatic incident with a cat) or (3) heard about a traumatic event with others, or that a cat is "dangerous".
Dealing with fear for the child cats (ailurophobia) is simple and straightforward. It is through exposure to a cat in a graduated fashion, while enjoying the child to use relaxation to overcome feelings of anxiety during exposure. You can start with photos of a cat (from a line drawing of a photo). So the child can practice abdominal breathing to relax while watching the line drawing ... until the child can watch without fear, then gradually work on the image using the same procedure.
Once this is achieved, you child with a great cat, which is tied at a distance and go through the same relaxation process. When the child gets used, and can happily see the cat, the cat moved a bit closer. All movements are closer in small steps and occurs only when the previous distance is reached without fear.
At this point you can give the child a stuffed animal version of the cat with that knowledge and feel confident. The child can also be taught how to do a visualization where you child can pet the cat and I feel how soft the fur is and how comforting and anxiety-reducing spinning. This may take a while because the visualization takes focus and concentration for your baby. But this move makes the process more fair and sensible.
Your child should also check other friends with the cat and receive a friendly reply. This can happen to an alternative negative experience with a positive replacement. If the cat was very moved close to the child in exposure-relaxation process, promote the child's pet cat as others see him / her how.
In most cases, fear of cats (or other animals), this will work if you exhibit patience and perseverance. Although this can also be used for adults with a similar fear can be especially successful with children who still have fluid in their learning, thinking and feeling, and this fear has not cemented itself as adults.
By addressing the child's cat anxiety early on, also you risk your cat is affected negatively by the child's phobic reaction to it, which will be a win-win.