Out of interest, I'm looking for the word describing the usual cap of detergent for manual dish-washing. It consists of two parts which can be moved relatively to each other in vertical orientation so that pull up opens the bottle and pushing down closes it. The bottle is then turned upside-down and the detergent can be dosed.
A soap dispenser [for liquid soap] is different from what I'm searching since it requires a push (usually vertical) to press out the next portion.
That being said, I'm only speculating that a specific word for it exists. I'd use dish-washing detergent cap if I'd be force to make a choice.
I stumbled over this before asking a question on chemistry.SE about the condensation rate of detergent in a bottle with such a cap, i.e. an example sentence is "What would be an appropriate estimate for the condensation rate of dish-washing detergent in a bottle closed with a X" (I'll clarify the context in the chemistry.SE question, of course). I'm aware that it's not necessary to know the word to ask the question there.
Best Answer
According to numerous sources, a cap of this design is a push pull cap, without a hyphen.