Can I calculate the checksum of a file as I create it?

I am creating a large tar archive and I would like to create the checksum of the archive too. I could achieve it like this:

$ tar cfz archive.tar.gz files
$ sha256sum archive.tar.gz > archive.tar.gz.sha256sum

But the archive file is huge and on slow media, so I'd prefer not to have to read it all in again after writing it out.

Can I build a pipeline that will hash the file as it writes it? I thought maybe I could do this with the tee utility, but that only writes to a file, not to the standard input of another command.

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