Why crawl property are automatically created for Lookup site columns, while Manged property will be missing

I am working on a team site collection, and inside this site collection i got 2 lists:-

1. Project
2. Departments2

Inside the Project list , i created a site column of type Lookup (name = ProjectDepartment) which is linked to the Department list, here is the ProjectDeparment column definition:-

now i created a new list, and i added the Project CT which contain the ProjectDepartment site column. I add some data inside the project & departments2 , and everything worked well. then i run a full crawl.

but i have noted that inside the Search Schema , there is not any managed property for the Lookup column (ProjectDeptarment), here is how the Managed Properties looks like:-

while seems i got a Crawl property for the ProjectDepartment , as follow:-

so i have these 2 questions:-

 - why i did not get any managed property for the Lookup column? and is there any problem if i manualy create it, and link ti to the related crawl property?

- If the answer tot the first question is Yes.. then can i do a search inside a content search webpart for all the Projects items that are linked to a specific Department Title ? something such as such as:-

ProjectDepartmentOWSTEXT:"DepartmentA"