Backbone has multiple options for synchronizing the component field.
Same component to same component
If you want to synchronize all issues to the same component value in both instances, simply select the Components Passthrough Mapping option.
If one synchronization partner tries to synchronize a component with an instance without a component with the same name, the synchronization won't work. To avoid this, select Create missing components to automatically create a component with the same name in the partner instance when the synchronization is initialized.
Specific component to specific component
You can synchronize specific components to other specific components (for example, you can synchronize issues in component '30' in Project A to component 'Thirty' in Project B).
To do so, select the Components Field Value Mapping setting, and in the Component mapping section, click Add Entry to enter the components you want to synchronize. The components are mapped according to the number they are assigned to – so the component assigned to number 1 on the left side will be synchronized to component 1 on the right side.
You must also select a default component for the issue to be mapped to in case you are synchronizing a component that you haven't created a mapping for. You can do this using the Default component dropdown.