Send alerts from loki via "prometheus remote write" to grafana

How can I forward alerts that are created by loki via “prometheus remote write” to grafana? loki is already able to send alerts via “prometheus remote write” to alertmanager.
grafana: 10.0.3
prometheus: 2.45.0
loki: 2.9.3

Not sure what you mean. Why do you want to send alerts to Grafana if you already have an alertmanager?

We do not have an 24/7 support. My colleagues want to have a nice dashboard when they start to work in the morning, showing the alerts of each server in a nice overview. I think that alertmanager does not provide such a dashboard?

Also I need to visualize an aggregation of the log data in grafana. The logs provide entries for each department connected. A dashboard shall show how active each department is and how the activity changes over time. Is this possible via loki and prometheus write?

You can have dashboards in Grafana and alerts in alertmanager, or you can have both alerts and dashboards in Grafana, both would be just fine. It depends on what your preference is and if you have any tooling around either solution.