Final answer:
The phrase "for running" is considered a prepositional phrase, which could modify either a verb, an adjective, or another adverb within a sentence.
Step-by-step explanation:
The phrase "for running" is a prepositional phrase. A prepositional phrase begins with a preposition and can function in several ways within a sentence, including as an adverbial or adjectival modifier. In this case, "for" is the preposition and "running" is a gerund, an -ing verb functioning as a noun. Together they make up the prepositional phrase which could act as an adverbial or adjectival modifier depending on its placement in a sentence.