Which 'up' is a preposition?

Up as an adverb:
• We use up as an adverb to talk about movement towards a higher position, value, number or level.
- She put the books up on the highest shelf. -- Adverb

Up as a preposition:
• We use up to talk about a higher position or movement to a higher position
- He was up a ladder painting. -- Preposition

Up as an adjective:
• We use up as an adjective usually to talk about increases in prices, levels or amounts
- The price of fuel is up again. -- Adjective
- It was cold yesterday but the temperature is up today. -- Adjective

Source: Cambridge Dictionary