long a cap is a position similar to buy a call on LIBOR while long a floor is a position similar to buy a put on LIBOR
第一句话是对的，因为cap本质上就是一系列的利率看涨期权，floor是一系列的利率看跌期权

可后面又说a long cap is equivalent to a portfolio of long put options on fixed-incomes securities and a long floor is equivalent to a portfolio of long call options on fixed-income securities prices.
long cap   利率上升获利，，而利率上升，fixed-income价格下降，则long put option获利，即两者在利率上升的时候，都是获利的。

The underlyings of the call and put are different.

For the first scenario as you mentioned, the underlying are LIBOR rate

For the second scenario, the underlying are the price of the fixed income securities