In this case, the word "designate" is being used as a noun. It is not being used as an adjective or verb.
For example we might write,
"The designate strode down the hallway" (Example 1)
Any English verb which ends in "ed", when written in the past-tense, can be used as a noun it you omit the "ed" suffix.
- "The work stood upon the dias like some sort of Greek God." (Example 2A).
The word "work" is being used as a noun.
"The cook was leaning against the wall int he alley, and smoking a cigarette, while gazing wistfully at the setting sun. (2B).
we went for a long walk. (Example
2C)
"a kiss upon the cheek would have made her feel better; but such displays of affection were not allowed in public." (Example 2D)
a look into the tank would have made most people's stomaches turn, but Susan was not a squeamish woman. (Example
2E). The word "look" is being used as a noun, it as an adjective.
"Gonzalez did a good job on the car repair"(2F)
Likewise, "designate" can be used as a noun.
Note than in the sentence, "1 + 2 = 3", it would incorrect to say that 1 is an adjective which modifies 2. It would also be incorrect to say that 2 is an adjective which modifies 1.
Likewise, in the phrase "manager designate" the words "manager" and "designate" have equal precedence.
In a syntax tree, parse tree, or other tree diagram, it would be poor practice to have "manager" be a child of "designate" or write "designate" as a child of "manager"
There is an invisible plus sign which gives us "manager + designate"
The phrase "manager designate" being used as if it was a noun.
I reccomend that you not always analyze English sentences one word at a time.
An entire phrase of two or more words can act as a noun or act as a verb.
Likewise "manager designate" can be assigned to one single node in a tree diagram. You can draw children underneath the node if you wish to further sub-divide things. There is a branch of mathematics which is somtimes useful in linguistics.
Tree diagrams are a powerful tool.