Floating exchange rate and cost-push inflation? I have a trouble understanding the following text from my textbook. I bolded the fragments that don't make sense to me.

"Disadvantages of a floating exchange rate:
A floating exchange rate regime may worsen existing levels of inflation. If a country has high inflation relative to other countries then this will make its exports less competitive and its imports relatively less expensive. The exchange rate will then fall, in order to rectify the situation. However, this could lead to even higher import prices of finished goods, componetns, and raw materials, and thus cost-push inflation, which may further fuel the overall inflation rate."

If there's high inflation, exports become less competitive and imports become less expensive - that's obvious. But why do they say that it would make imports' prices even higher? Why "even" and why "higher"? They've said that imports are cheaper after inflation, so imported raw materials, etc. should be cheaper as well so the costs of productions should be cheaper too and don't cause cost-push inflation but totally opposite.
I would imagine that a falling exchange rate causes the price of imports to rise as the foreign currency becomes stronger against the local currency.

With high inflation exports become more expensive because foreign buyers have to exchange more of their currency to account for the increase in price. This means they cost more per unit in terms of the foreign currency and consequently become less competitive.

The subsequent fall in demand for exports causes the fall in the exchange rate as the demand for exports and local currency are complementary. A weaker local currency means that one unit of local currency is now worth fewer units of foreign currency.

Thus the result is that imports become more expensive as any given amount of imports now costs more in terms of the local currency. Imagine that the exporting nation, say England, sells teddy bears for £5.00 a piece. If the dollar weakened against the pound from \$1.50 to a pound to \$1.56, then as far as a buyer from the United States is concerned, the price of a teddy bear increases from \$7.5 (i.e. 1.50 * 5.00) to \$7.8 (i.e. 1.56 * 5.00).