There's no compiler here - it's a tool for bidirectionally synchronizing files (specifically code, assets, build products, etc.) and forwarding network traffic to and from remote systems.

The idea behind Mutagen is to use your local tools (e.g. editor, IDE, browser, etc.) to work on remote hardware (where "remote" can mean anything you like, e.g. your local system, a Docker container (local or remote), a Raspberry Pi, a cloud server, etc. - or some combination of all of those).

The architecture is designed to be maximally flexible, so you can work between your local system and a remote system, between two remote systems, or in any other topology you want.

In the kind of setup you'd use with mutagen your dev tools and compilers are typically inside the container, while your code and editor are outside the container and being synced inside the container on every change. If the container is running on another machine/cloud service then yeah it could be that your compiler is running outside your machine. But much more likely it's still running as a container on your machine. The typical use case for this is making your dev environment accessible to anyone, just send them a dockerfile and they can almost perfectly reproduce it and get going with building your software (vs. having to laboriously setup the exact version of each dev tool and compiler you use).