Python aiohttp module: ambiguous .content attribute

Here is a little code snippet:

import aiohttp
import aiofiles

async def fetch(url):
    # starting a session
    async with aiohttp.ClientSession() as session:
        # starting a get request
        async with session.get(url) as response:
            # getting response content
            content = await response.content
            return content
 
async def save_file(file_name, content):
    async with aiofiles.open(f'./binary/{file_name}', 'wb') as f:
      while True:
            chunk = content.read(1024)
            if not chunk:
                break
            f.write(chunk)

I am trying to download some binary files using the aiohttp library and then passing them to a coroutine using aiofiles library to write the file in the disk. I have read the documentation but still couldn't figure out if I can pass content = await response.content or is it closed when the handle async with.. is closed? Because on a secondary blog, I found:

According to aiohttp’s documentation, because the response object was created in a context manager, it technically calls release() implicitly.

Which confuses me, should I embed the logic of the second function inside the response handle or is my logic correct?

Thanks to user4815162342's code I could find a solution by parellelizing the fetch and write coroutines. I would've checked his code as the accepted solution but since I had to add some code to make it work, here it is:

# fetch binary from server
async def fetch(url):
    async with aiohttp.ClientSession() as session:
        async with session.get(url) as response:
            async for chunk in response.content.iter_chunked(4096):
                yield chunk

# write binary function
async def save_file(file_name, chunk_iter):
    list(map(create_dir_tree, list_binary_sub_dirs))
    async with aiofiles.open(f'./binary/bin_ts/{file_name}', 'wb') as f:
        async for chunk in chunk_iter:
            await f.write(chunk)
    

async def main(urls):
    tasks = []
    for url in urls:
        print('running on sublist')
        file_name = url.rpartition('/')[-1]
        request_ts = fetch(url)
        tasks.append(save_file(file_name, request_ts))
    await asyncio.gather(*tasks)

asyncio.run(main(some_list_of_urls))