It seems that the issue you're facing is related to using Inertia.js with Laravel. In Inertia.js, you cannot return a traditional redirect response from your controller. Instead, you should use Inertia's native methods to handle redirection.

Update your authenticateAgency method to return an Inertia response:

use Inertia\Inertia;

public function authenticateAgency(Request $request)
{
    if (Auth::guard('agency')->attempt($request->only('email', 'password'))) {
        $request->session()->regenerate();
        return Inertia::location(route('agencyHome'));
    } else {
        Auth::logout();
        return Inertia::render('YourLoginFormComponent', [
            'error' => 'You are not authorized to access this resource',
        ])->withInput($request->only('email'));
    }
}

Replace YourLoginFormComponent with the name of your Vue.js login form component.

In your Vue.js login form component, make sure to handle and display the error message:

<template>
  <!-- Your form markup -->
  <div v-if="error">{{ error }}</div>
</template>

<script>
export default {
  props: {
    error: String,
  },
  // rest of your component code
};
</script>

By using Inertia's native methods, you should be able to handle redirection and display error messages in your Laravel + Inertia.js application.