How to assign values from one object to another

Your code doesn't explain the main problem, I see some errors in that piece of code, I can't write a precise answer but I can help you to decrease your errors:

Indentation is important:

async componentDidMount() {
  try {
    const array = [
      {
        expanded: false,
        category_Name: "Mobiles",
        sub_Category: [
          { id: 1, name: 'Mi' },
          { id: 2, name: 'RealMe' },
          { id: 3, name: 'Samsung' },
          { id: 4, name: 'Infinix' },
        ],
      },
      {
        expanded: false,
        category_Name: "Laptops",
        sub_Category: [
          { id: 8, name: 'Dell' },
          { id: 9, name: 'MAC' },
          { id: 10, name: 'HP' },
          { id: 11, name: 'ASUS' },
        ],
      }
    ];
    // What listTodos mean?
    const graphqldata = await API.graphql(graphqlOperation(listTodos));

    console.log('graphqldata:', graphqldata)
    //  Unnecesary
    // this.setState({ AccordionData: array });

    // You need to pass a function to run after the state changes
    this.setState({
      AccordionData: array,
      todos: graphqldata.data.listTodos.items,
    }, () => {
      this.state.todos.map((item) => { // You forgot to wrap your map function
        console.log("printing just name", item.name)
        const singleAccordion = {}; // Better than new Object()
        singleAccordion.category_Name = item.name;
        singleAccordion.expanded = false;
        singleAccordion.sub_Category = [
          { id: 1, name: item.date },
          { id: 2, name: item.location },
          { id: 3, name: item.organizer },
        ];
        console.log("AND PRINTING singleAccordion", singleAccordion)
      });
    })
    // This isn't available till state has changed
    console.log('AccordionData:', this.state.AccordionData)
  } catch (err) {
    console.log('error fetching data: ', err)
  }
}

Hope it helps.

EDIT by question OP: Here's the rest of code and the render method:

 class AboutScreen extends React.Component {
 render(){
  return(
    <View style={styles.Panel_Holder}>
    <Text style={styles.category_Text}>About Screen </Text>
    <Text style={styles.sub_Category_Textt}>Put a mission statement 
    </Text>
   <Button
      title="Go to Home"
      onPress={() => this.props.navigation.navigate('Home')}
   />
   </View>

  )
 }
 }

  class Expandable_ListView extends Component {

  constructor() {

  super();

  this.state = {

  layout_Height: 0

  }
  }

 componentWillReceiveProps(nextProps) {
  if (nextProps.item.expanded) {
     this.setState(() => {
     return {
      layout_Height: null
     }
   });
  }
  else {
    this.setState(() => {
      return {
        layout_Height: 0
      }
      });
   }
   }

   shouldComponentUpdate(nextProps, nextState) {
   if (this.state.layout_Height !== nextState.layout_Height) {
     return true;
   }
   return false;
  }

  show_Selected_Category = (item) => {
  Alert.alert(item);

  }

 render() {
 return (
  <View style={styles.Panel_Holder}>

    <TouchableOpacity activeOpacity={0.8} onPress= . 
    {this.props.onClickFunction} style={styles.category_View}>

      <Text style={styles.category_Text}> . 
     {this.props.item.category_Name} </Text>

      <Image
        source={{ uri: 'https://reactnativecode.com/wp- 
        content/uploads/2019/02/arrow_right_icon.png' }}
        style={styles.iconStyle} />

    </TouchableOpacity>

    <View style={{ height: this.state.layout_Height, overflow: 
    'hidden' }}>

      {
        this.props.item.sub_Category.map((item, key) => (

          <TouchableOpacity key={key} style= . 
       {styles.sub_Category_Text} onPress= . 
       {this.show_Selected_Category.bind(this, item.name)}>

            <Text> {item.name} </Text>

            <View style={{ width: '100%', height: 1, backgroundColor: 
          '#000' }} />

          </TouchableOpacity>

        ))
       }

      </View>

    </View>

   );
  }
  }

  class HomeScreen extends React.PureComponent {

   state = { isloading: true, name: '', date: '', location: '', 
   organizer: '', sections:'', link:'', description:'', todos: [], 
   AccordionData: [], AccordianPlus:[] }


   async componentDidMount () {
   try {
     const array = [

    {
      expanded: false, category_Name: "Mobiles", sub_Category: [{ id: 
     1, name: 'Mi' }, { id: 2, name: 'RealMe' }, { id: 3, name: 
    'Samsung' },
    { id: 4, name: 'Infinix' }, { id: 5, name: 'Oppo' }, { id: 6, 
    name: 'Apple' }, { id: 7, name: 'Honor' }]
    }
    ];

   const graphqldata =  await 
       API.graphql(graphqlOperation(listTodos))

   console.log('graphqldata:', graphqldata)

   this.setState({
    isloading: false,
    AccordionData: [],
    AccordianPlus: [],
    todos: graphqldata.data.listTodos.items,
    }, () => {
     this.state.todos.map((item) => { // You forgot to wrap your map 
    function
    console.log("printing just name", item.name)
    const singleAccordion = {}; // Better than new Object()
    singleAccordion.category_Name = item.name;
    singleAccordion.expanded = false;
    singleAccordion.sub_Category = [
      { id: 1, name: item.date },
      { id: 2, name: item.location },
      { id: 3, name: item.organizer },
    ];
    console.log("AND PRINTING singleAccordion", singleAccordion)
    this.state.AccordianPlus.push(singleAccordion)

  });
})
 console.log('AccordionData:', this.state.AccordionData)
 console.log("Did AccordionPlus work?", this.state.AccordianPlus)
 this.state.AccordionData.push(this.state.AccordianPlus)
 console.log("Did final concat work?", this.state.AccordionData)
 } catch (err) {
  console.log('error fetching data: ', err)
 }
 }

 update_Layout = (index) => {

LayoutAnimation.configureNext(LayoutAnimation.Presets.easeInEaseOut);

const array = [...this.state.AccordionData];

array[index]['expanded'] = !array[index]['expanded'];

this.setState(() => {
return {
  AccordionData: array
}
});
}

render() {

if (this.state.isloading){
   return <View><Text>Loading...</Text></View>;
 }
return (
  <View style={styles.MainContainer}>
  <Button
    title="About"
    onPress={() => this.props.navigation.navigate('About')}
  />
  <Text style={styles.Title_Text} > Hello Chess Finder </Text>
  <ScrollView contentContainerStyle={{ paddingHorizontal: 10, 
   paddingVertical: 5 }}>
    {
      this.state.AccordionData.map((item, key) =>
        (
          <Expandable_ListView key={item.category_Name} 
        onClickFunction={this.update_Layout.bind(this, key)} item= . 
     {item} />
        ))
    }
  </ScrollView>

  </View>
  );
 }
 }