The same sentence can be made bigger by being built on to. The cat itself can be described adjectivally:
The white cat, the ginger cat, the Siamese cat, or the verbal function of the cat can be developed adverbially:
The cat is sitting on the mat.
‘On the mat’ is an adverbial phrase (phrase because it contains no finite verb) showing where the cat sits.
More can now be said about the cat verbally by adding another verb joined by and, but, or.
The cat is sitting on the mat and is sleeping.
This gives a compound sentence. The same idea could be conveyed adverbially:
The cat sits sleeping (or asleep) on the mat.
It could also be conveyed adjectivally:
The sleeping cat is sitting on the mat.
or,
The sitting cat is sleeping on the mat.