I want to store the details of the user in a collection in my firestore database when I signup with the flutter fire UI auth package. so I can update the user details when I click on the profile screen

Here is my code:

import 'package:firebase_auth/firebase_auth.dart';
import 'package:flutter/material.dart';
import 'package:flutterfire_ui/auth.dart';
import 'package:ghibliapp/src/screens/home_screen.dart';
import 'package:lottie/lottie.dart';

class AuthGate extends StatelessWidget {
  const AuthGate({super.key});

  @override
  Widget build(BuildContext context) {
    return StreamBuilder<User?>(
        stream: FirebaseAuth.instance.authStateChanges(),
        builder: (context, snapshot) {
          if (!snapshot.hasData) {
            return SignInScreen(
              showAuthActionSwitch: true,
              subtitleBuilder: (context, action) {
                return Padding(
                  padding: const EdgeInsets.only(bottom: 8),
                  child: Text(
                    action == AuthAction.signIn
                        ? 'Welcome to Ghibli! Please sign in to continue.'
                        : 'Welcome to Ghibli! Please create an account to continue',
                  ),
                );
              },
              footerBuilder: (context, action) {
                return Padding(
                  padding: const EdgeInsets.only(top: 16),
                  child: Text(
                    action == AuthAction.signIn
                        ? 'By signing in, you agree to our terms and conditions.'
                        : 'By signing up, you agree to our terms and conditions.',
                    style: const TextStyle(color: Colors.black),
                  ),
                );
              },
              headerBuilder: (context, constraints, _) {
                return Padding(
                  padding: const EdgeInsets.all(20),
                  child: AspectRatio(aspectRatio: 1, child: LottieBuilder.asset("lib/assets/splash.json"),),
                );
              },
              providerConfigs: const [
                EmailProviderConfiguration(),
              ],
            );
          }
          return const HomeScreen();
        });
  }
}

Any help would be appreciated.