Firebase Functions V2 - Use custom domain for http calls [duplicate]

I'm using Firebase functions v2 and I would like to use a custom domain name for HTTP calls. I already searched before posting but all answers are for Firebase functions v1.


Here is the function showed in the Firebase console:

https://-ltnwgqwdfq-uc.a.run.app />


And here is what I would like to achieve:

https://.my-domain.com />


Here is what I tried:


In firebase.json file
{
"hosting": {
"rewrites": [
{
"source": "/api/function-name",
"function": {
"functionId": "function-name"
},
{
"source": "!/@(api)/**",
"destination": "/index.html"
}
]
}
// [...]



I deployed only the hosting part and when I try to run the google cloud function it works well:

https://function-name-ltnwgqwdfq-uc.a.run.app />


But impossible to run the custom domain name function (it redirects me to a 404 HTML page):

https://my-domain.com/function-name