Convert to nested object using Object.entries

How can I format an object with a basic structure to a complex structure containing arrays in an efficient way?

Sample input:

const obj = {
  OBJ1: "obj1",
  OBJ2: "obj2",
}

const mapper = {
   "obj1": "Object1"
   "obj2": "Object2"
}

Expected output:

const converted = {
 obj1 : { name: "Object1", arr: [] }
 obj2:  { name: "Object2", arr: [] }
}

I need to run a map on key value pairs of object obj, and to that key I need to add a property called name with its value from another object mapper and then add arr array as another property to this key

I tried:

const convert = () => {
  const newObj = {};
  Object.entries(columnsToInclude).map(
    ([key, value]) => (newObj[key] = { name: mapper[value], arr: [] })
  );
};