同僚 and 仲間: coworkers and companions in an activity
Distinguish sharing a workplace from sharing an activity.
Coworkers
同僚 (どうりょう) means a coworker. 「同僚と昼食を食べました」 says the speaker had lunch with a coworker.
「同僚に資料を渡しました」 means handing materials to a coworker. 同僚 alone does not establish a supervisor-subordinate relationship.
A shared activity
仲間 (なかま) refers to companions sharing an activity or purpose. 「山登りの仲間と話しました」 means talking with hiking companions.
「仲間に入れてください」 asks to join the group or activity. It differs from simply entering a building.
Relationships can overlap
A coworker may also be a 仲間 in a shared activity. The two words do not name mutually exclusive categories.
「同僚ではありませんが、音楽の仲間です」 says the person is not a coworker but a companion in music. It changes the basis used to describe the relationship.
Must 「山登りの仲間」 work at the same company?
Show the answer and why
No. It identifies hiking as the shared activity, not a shared employer.