How to check if the JSON Object array contains the value defined in an arrary or not?

Try the following:

var categories = [
    {catValue:1, catName: 'Arts, crafts, and collectibles'},
    {catValue:2, catName: 'Baby'},
    {catValue:3, catName: 'Beauty and fragrances'},
    {catValue:4, catName: 'Books and magazines'},
    {catValue:5, catName: 'Business to business'},
    {catValue:6, catName: 'Clothing, accessories, and shoes'},
    {catValue:7, catName: 'Antiques'},
    {catValue:8, catName: 'Art and craft supplies'},
    {catValue:9, catName: 'Art dealers and galleries'},
    {catValue:10, catName: 'Camera and photographic supplies'},
    {catValue:11, catName: 'Digital art'},
    {catValue:12, catName: 'Memorabilia'}
];

var categoriesJson = JSON.stringify(categories);
var mainCat = ['Arts, crafts, and collectibles', 'Baby' , 'Antiques']

$.each(JSON.parse(categoriesJson) , function (key, value) {
  if(mainCat.indexOf(value.catName) > -1){
   console.log('Exists: ' +value.catName)
 }
 else{
   console.log('Does not exists: ' +value.catName)
 }
});
<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>


I would filter the initial data array to get the array of matching categories:

var matchedCategories = categories.filter(i => mainCat.indexOf(i.catName) >= 0);

Then you could do what you need by iterating this sub-array.