Are the underlined determiners correctly used?

We had our lunch there and enjoyed ourselves very much. Unfortunately when we began to walk down the mountain, we were caught in a storm. Faced with the unexpected situation, our teacher remained calm, asking us to help each other down.
The original text:
"Today we went to climb Xiangshan Mountain.
Early in the morning we met at the school gate and took a bus there. At nine o’clock we got to the foot of the mountain and started to climb. The mountain was very high and some of us got tired soon, but all of us were very excited. We cheered happily when we reached the top. The view was wonderful.
We had our lunch there and enjoyed ourselves very much. Unfortunately when we began to walk down the mountain, we were caught in a storm. Faced with the unexpected situation, our teacher remained calm, asking us to help each other down.