This was carried sharply down through the skin and subcutaneous tissue to the transverse carpal ligament.

Identify Parts of Speech:

This = adjective
sharpley = adverb
through = preposition
and = conjunction
tissue = noun
to = preposition
transverse = adjective
ligament = noun

Correct, yes.

Technically, the first one is sometimes called a pronominal adjective, but not always! That's fairly pedantic!

Correct identification of the parts of speech! Here is a breakdown:

This - adjective, used to modify and describe a noun (carried)
sharply - adverb, used to modify and describe the verb (carried)
down - preposition, indicating direction or movement
through - preposition, indicating movement from one side or end to another
the - article, used to specify a particular noun (tissue)
skin - noun, the outer covering of the body
and - conjunction, used to connect words or phrases
subcutaneous - adjective, describing the tissue beneath the skin
tissue - noun, a group of cells that perform a similar function
to - preposition, indicating movement or direction
the - article, used to specify a particular noun (transverse carpal ligament)
transverse - adjective, describing the carpal ligament
carpal - adjective, describing the ligament
ligament - noun, a band of strong tissue that connects bones or supports organs