AWS Lambda successfully called with 'Function URL', but not with 'API Gateway'

Question

I am new to AWS. I'm trying to use Lambda to deploy a backend made with FastAPI and Mangum.

It works well if I use Lambda's 'Function URL', but if I use the 'AWS API Gateway' URL, it seems that somehow the path is not resolved correctly. What am I doing wrong?

I tested it with code similar to the following:

from fastapi import FastAPI 
from mangum import Mangum

app = FastAPI()
# app = FastAPI(root_path = '/default_stage/myFunction')

handler = Mangum(app, lifespan="off")
# handler = Mangum(app, lifespan="off", api_gateway_base_path='/default_stage/myFunction')

@app.get("")
def f1():
    return "f1"
@app.get("/")
def f2():
    return "f2"
@app.get("/k")
def f3():
    return "f3"

@app.get("/myFunction")
def f4():
    return "f4"
@app.get("/myFunction/")
def f5():
    return "f5"
@app.get("/myFunction/k")
def f6():
    return "f6"

@app.get("/default_stage/myFunction")
def f7():
    return "f7"
@app.get("/default_stage/myFunction/")
def f8():
    return "f8"
@app.get("/default_stage/myFunction/k")
def f9():
    return "f9"

And I tried FastAPI(root_path = '/default_stage/myFunction') or Mangum(app, lifespan="off", api_gateway_base_path='/default_stage/myFunction') and somehow it seems to be related to the solution, but I couldn't figure it out.

Also tried changing API Gateway > Develop > Routes here and there, but this didn't work either.

Any advice would be greatly appreciated, thanks.