It is poor practice to vent under the eaves because then the exhaust goes up through the soffits and into your attic space which can create bad vapour problems. When we had our house done, the HVAC guy ran a duct from the kitchen to the gable end through the attic. This had the added benefit of allowing us to hook the bathroom and kitchen vents all to the same duct.

You need to increas the CFM when you have either long runs or sharp turns

Thank you. No worries about the eaves as this will be directly outside. There is only one turn and the distance is as seen on the photo. GN, I was hoping for a stronger unit. What is "make up air"?

What can I expect to pay for this job?

Makeup air is when you have to provide an air supply to the kitchen, here is 1 example, but if you plan on doing that call HVAC company to do the duct work and they will give you the options how to provide make up air. Under 400CFM you can get appliance store installers do the work, just make sure they licensed and insured.

That you would have to get some local prices, if one person does everything, it will be around 750-1k give or take. Just make sure they rough everything in before you install cabinets.

We get calls sometimes to install the hood vent and the kitchen is already installed and it's always the same excuse "originally it was supposed to be a ventless fan"

Good luck

Make-up Air (MUA)

All the air that actually is pulled into the hood and removed to the outside has to come from somewhere, and none of the blowers used with hoods can suck the house pressure down to a vacuum. Hence, the air that goes out is made up by new air flowing into the house.

When the air that goes out is very modest in flow rate (CFM), house leakage may suffice to replace it. Otherwise, a deliberate means of replacing the air is needed. This means is called the MUA system. In concept, it is like the hood ventilation system, only in reverse. It may require a method of heating the air, depending on locale.

Safety, among other issues requires that the house pressure not drop by
much. This is why MUA is essential when higher flow rate hood
ventilation systems are used.

So, no air leaves via the hood without being replaced. If the replacement path is minimal (house leakage) the hood will not even pull 2/3 of its blower's rated flow rate. With MUA, 2/3 is possible. Many messages are available on this site describing how much CFM you should have for cooking plume containment by the hood, and the size of the hood needed for full capture.

Just reading the last 20 or so hood threads should provide sufficient insight that you will recognize some of the issues involved in kitchen ventilation.

P.S. In the nice figure provided by GN Builders, the MUA register should be a diffuser aimed down the kitchen away from the hood so that the MUA doesn't disturb the rising cooking plumes. Hoods cannot (at tolerable noise and air flow levels) suck cooking effluent from the stove top. They depend on the rising hot plumes natural upward velocity to get the effluent into the hood entry aperture. Any cross drafts, or disruption of the plume shape is likely to degrade capture efficiency.

Thank you so much for your time!