Mate is built to be used from inside other apps: Mail, Messages, Notes, Pages, Word, Slack, a PDF, a terminal. You never need to copy text into a translator window by hand. Three ways, from most to least automatic.
1. Right-click › Services
Select the text, right-click, and choose Services › Translate with Mate Translate. Mate opens with the translation. This is available in every native Mac app that uses the standard text system, which is nearly all of them.
If you use it a lot, give it a keyboard shortcut: System Settings › Keyboard › Keyboard Shortcuts › Services › Text › Translate with Mate Translate. Then it's one keystroke from any app.
2. Copy and Option+Shift+T
Turn on Auto-translate clipboard in Mate's Settings › General. From then on, Cmd+C followed by Option+Shift+T opens Mate with the copied text already translated. This works even in apps that replace the context menu, such as some PDF viewers, Electron apps and remote desktops.
There's also a Translate clipboard button in the window for when the setting is off.
3. The menu bar
Click the Mate icon in the menu bar, or press Option+Shift+T, and type or paste. Mate keeps your last translation there.
Getting the translation back into the app
Turn on Automatically copy translation in Settings › General and every translation lands in the clipboard, ready to paste with Cmd+V. Or select the translation in Mate and copy it.
In Safari
Safari has its own, nicer integration: the floating button and double-click. See How to use Mate's Safari extension on Mac.
From scripts and other apps
Mate accepts URL links, for example matetranslate://translate?text=hello&from=en&to=de. The list is in Mate's Settings › Deep Links. Automation tools such as Shortcuts, Raycast and Alfred can open those links.