"E-mail address already been taken" is responsibility of validator library or UI?

Le'ts suppouse we have the follow architrecture..

Lib (npm library) that validate some data:

So for instance.. the RegisterForm have the fields (Name, email and password) The RegisterForm imports the library (validator) that validate the data (name, email and password)

import registerValidator from 'validator'

onSubmit(values) {
    if (registerValidator.validate(values) {
        // Is valid, we can register new user...
    }
}

One of the rules that should be follow before the user can be registered, is that the e-mail address should be unique on our database.

The question is:

1. should I made this validation (fetch my API to check if the e-mail was already taken) on validator library ?
2. should I validade this on onSubmit after the data is valid.. something like this.

This...

import registerValidator from 'validator'

onSubmit(values) {
    if (registerValidator.validate(values) { // perform fetch operation to check on our database if the email was already taken
        // Is valid, we can register new user...
    }
}

Or this ...

import registerValidator from 'validator'
import isUniqueEmail from 'api/user';

async onSubmit(values) {
    if (registerValidator.validate(values) { // Is valid e-mail
        const isUniqueEmail = await isUniqueEmail(values.email); // is not taken yet
        if (isUniqueEmail) {
            // Is valid, we can register new user...
        }
    }
}

I know that the post method for the user register should check this .. but this is only an example to ask where the rule `e-mail must be uniqueà should placed