API-gateway-lambda : Cookie not getting saved in browser - cookies

Question

I'm trying to know how to use cookies with AWS-Lambda with the serverless framework as per this blogpost
and following is my serverless.yml code
functions:
hello:
handler: handler.hello
events:
- http:
path: /post
method: post
cors:
origin : 'https://xyz.netlify.app'
and Lambda function as per following
"use strict";
const cookie = require("cookie");
module.exports.hello = async (event) => {
const body = JSON.parse(event.body);
const name = body.name;
const value = body.value;
return {
statusCode: 200,
headers: {
"Access-Control-Allow-Origin": "https://xyz.netlify.app",
"Access-Control-Allow-Credentials": true,
"Set-Cookie": cookie.serialize(name, value, {
expires: new Date(new Date().getTime() + 10 * 1000),
}),
},
body: JSON.stringify(
{
input: event,
},
null,
2
),
};
// Use this code if you don't use the http event with the LAMBDA-PROXY integration
// return { message: 'Go Serverless v1.0! Your function executed successfully!', event };
};
As you can notice, I already have configured the code to avoid any cors issue.
While try to send a post request as per following,
const name = document.getElementById('name')
const value = document.getElementById('value')
const post_btn = document.getElementById('post_btn')
post_btn.addEventListener('click', () => {
console.log(name.value, value.value)
const post_url = 'https://abcdxyz59t9.execute-api.ap-south-1.amazonaws.com/dev/post'
const user = {
name: name.value,
value: value.value
};
// request options
const options = {
method: 'POST',
body: JSON.stringify(user),
headers: {
'Content-Type': 'application/json'
}
}
// send POST request
fetch(post_url, options)
.then(res => res.json())
.then(res => console.log(res));
})
I do get a Set-Cookie header like below
But the cookie doesn't get saved in the browser.
That's not the case when I directly try to hit a get request with that URL without the cors in the browser. Can anyone please tell me what to do?