We have supply driven inflation as people were paid to do nothing or not produce as much.
We have supply driven inflation because people are buying things. It comes from a number of factors. There are still COVID initiated supply side problems. Just ask Ford about their Blue Oval badge shortage. There is also still pent-up demand from the serious supply side shortages over the last couple years.
And from a US perspective, we have been at full employment for quite a while and that is creating supply side inflation in the labor market. You may not get a raise from your current employer, but if you change jobs you likely will see a considerable increase.
Then of course, there is the steep increase in energy costs. Gasoline costs may have come down, but the diesel that gets all the goods to market is still very high.
The way the Fed 'cools off the economy' is by making things more expensive. They just want to choose where those more expensive items are. Here are two ways how it works: higher mortgages reduces the demand for new houses. Fewer new houses get built, fewer salaries required to build them. Labor market softens and consumer purchasing power starts to drop. Trickle down starts with the durable goods and goes from there. This ripples through the economy and the Fed hopes for a 'soft landing' when equilibrium shifts to less demand.
Then comes the second part, higher capital costs make it less profitable for businesses to expand. That softens labor demand. If product demand stays high they can raise prices to increase profits. But also costs increase the closer a business gets to maximum capacity with their current equipment.
So with fighting inflation its; be careful what you wish for. All paths leading to less inflation mean people are going to lose their jobs. Those who keep their jobs will be better off. Those who don't, well, it sucks to be you.