Not immediately.

> we recommend that you submit a notification pursuant to the DMCA directly to the originating site. In order for a DMCA takedown notice to a system caching service provider to be effective under 17 U.S.C. 512(b), the notice provider must demonstrate either (a) that the content has already been removed from the originating site or (b) the notice provider has obtained a court order requiring the content be removed from the originating site or access to the content to be disabled.

https://www.fastly.com/acceptable-use/