Zendesk Support and Zendesk Chat are two different applications and thus you'll need to install Tymeshift in each one.
A question we sometimes hear is: "Why do I need to install an app for Chat but not Talk?"
The reason for this is that Zendesk Talk is built into Zendesk Support, and thus, the Zendesk Support app automatically tracks Talk related activity. Since Zendesk Chat is a separate application and not housed within Zendesk Support, this requires the installation of a separate Tymeshift Chat app.
Here are setup instructions for each app: