what is the small hose that carries coolant to the intake?

right now it's 2 hoses of different sizes clamped and wired together so that it wont blow off and spray coolant every where. does this hose need a check valve of some kind? does it cool the injector in the intake or is it for temperature sensing? I dont even know what to ask for at the parts store.

I believe it's called the bypass hose, it supplies coolant to the manifold to keep it cool, since it's a big block of aluminium (usually) it needs cooling as it gets heat soak from the engine itself.
Hot inlet= low pressure on the compression stroke=lowered power from the burn.
Hence intercoolers on turbo cars.
Some of the other Mazda's use water for idle control and as a auto choke on cold mornings.