Question

My best friend and I enjoy wandering through the catacombs.

Answer 1

My best friend and I enjoy wandering through the catacombs. 
If the underlined verbal phrase is wandering through the catacombs, then it serves the role of object of the verb.
What do my friend and I enjoy? Wandering through the catacombs. 

Answer 2

Answer:

object of the verb through the catacombs.

Explanation:

Because if you ask to the verb where do you and your friend enjoy wandering?

the object to answer to this verb is the through the catacombs.