Monetary Policy - How can increasing interest rates cause higher inflation in the short term? My understanding of the interest rates part of monetary policy is the follows:

Borrower's point of view:

An increase in interest rates, increases the cost of borrowing, which may reduce disposable income, causing a reductiong in aggregate demand. The equillibrium between supply and demand will therefore need to be adjusted resulting in a lower average prices (less inflation).

Saver's point of view:

As interest rates rise, the reward for saving increases, therefore people are incentivised to save their money, which may cause a decrease in disposable income and less inflation.

However, I have a question in my textbook which says how can a rise in interest rates cause higher inflation in the short term, and I'm not sure.
If mortgage payments are included in the basket of goods that are used to calculate inflation, then rising interest rates will increase mortgage payments, causing an increase in inflation in the short term. This happens before any of the disinflationary effects show up in the data.