Hoping to understand why “fat” is used as an adjective and noun rather than “fatty” (at least in everyday English).

For example:

• “cat fat” (noun) and “fat cat” (adjective).
• “That is a fat cat”.
• Not “That is a fatty cat”.

Conversely:

• “cat skin” (noun) and ”skinny cat” (adjective).
• “That is a skinny cat”.
• Not “That is a skin cat”.
• Also not a “hair cat”, “fur cat”, “scare cat”, “smell cat” &c.

Also consider the the verb form:

• “there is more than way to skin a cat”.
• one would “fatten a cat” and not “fat a cat”
• you can also “scare a cat” etc.