Licensing
kutl is open source under two licenses. Almost everything is permissive; the relay is copyleft. This page is the canonical statement of which is which.
The split
The engine, the daemon, the CLI, the desktop app and the supporting libraries are dual licensed MIT OR Apache-2.0, at your choice: every crate in the open-source tree except the relay.
The relay, kutl-relay, is AGPL-3.0-only. It is the only server binary in the tree, and the only component that could be run as a competing hosted service, which is why it carries the stronger license.
What that means in practice
Building on the client side puts you under the permissive licenses. Embedding the sync engine, shipping a product that runs the daemon, or writing your own client against the protocol are all covered by MIT or Apache-2.0, whichever you prefer.
Running the relay as it ships, on your own infrastructure and for your own team, asks nothing further of you. See Self-hosting a relay. The AGPL terms matter when you modify the relay and offer it to others over a network: in that case the modified source has to be available to its users. This is a description, not legal advice. Read the license text for the terms that bind.
The license texts
All three live at the root of the published repository: LICENSE-MIT, LICENSE-APACHE and LICENSE-AGPL. Each crate also declares its own license in its Cargo.toml, which is the authoritative answer for that crate.
The hosted product
kutlhub is a commercial hosted service built on kutl. It is not open source, and nothing on this page applies to it. The protocol it speaks is the one documented here, and a space on kutlhub can be synced with the open-source client like any other.