Update user in database on success from Stripe prebuilt checkout

I am using Stripe's prebuilt checkout with react and firebase. The checkout process works fine and directs the user to the succes_url, but I would like to update a field under the user in the database as well. I don't understand how I can include a function that updates the DB that runs upon a successful checkout.

export const checkoutWithStripe = async(user) => {
    const checkoutSessionsRef = collection(db, 'users', user.uid, 'checkout_sessions');

    const singleCheckoutSessionRef = await addDoc(checkoutSessionsRef, {
        price: 'price_xyz',
        allow_promotion_codes: true,
        success_url: `${window.location.origin}/dashboard/app?success=true`,
        cancel_url: `${window.location.origin}/dashboard/app?canceled=true`
    });

    onSnapshot(singleCheckoutSessionRef, (snap) => {
        const { error, url: checkoutUrl } = snap.data();

        if (error) {
            console.error(`An checkout error occured: ${error.message}`);
        }
        if (checkoutUrl) {
            window.location.assign(checkoutUrl);
        }
        
    });

    // TODO: Update user type in firebase from free to starter on successful checkout
};

Thankful for any help.

Update:

I solved it, in 2 parts.

1. In Stripe I created a new webhook that points to a exported firebase function (2) that fires when "checkout.session.completed" is fired.

2. In Firebase i created a function that listens for the "checkout.session.completed" event and then calls a function that updates the DB based on the user email that I get from the Stripe event.

This is the Firebase function that listens to the event:

/**
 * A webhook handler function for the relevant Stripe events.
 */
exports.updateCustomer = functions.https.onRequest((req, resp) => {

    functions.logger.log("updateCustomer body", req);

    const relevantEvents = new Set([
        'checkout.session.completed'
    ]);
    let event;

    // Instead of getting the `Stripe.Event`
    // object directly from `req.body`,
    // use the Stripe webhooks API to make sure
    // this webhook call came from a trusted source
    try {
        event = stripe.webhooks.constructEvent(
            req.rawBody,
            req.headers['stripe-signature'],
            endpointSecret
        );
    } catch (error) {
        functions.logger.log(`Webhook Error: Invalid Secret`);
        resp.status(401).send('Webhook Error: Invalid Secret');
        return;
    }

    functions.logger.log("updateCustomer", event.type);

    if (relevantEvents.has(event.type)) {
        // logs.startWebhookEventProcessing(event.id, event.type);
        try {
            switch (event.type) {
                case 'checkout.session.completed':
                    const session = event.data.object;
                    functions.logger.log("checkout.session.completed:", session);
                    updatePlan(session.customer_details.email);
                    break;
                default:
                    functions.logger.log(`Unhandled event type ${event.type}`);
            }
        } catch (error) {
            functions.logger.log(`Unhandled event error ${event.type}`);
            resp.json({
                error: 'Webhook handler failed. View function logs in Firebase.',
            });
            return;
        }
    }

    // Return a response to Stripe to acknowledge receipt of the event.
    resp.json({ received: true });
});