1. write sth for 的意思是“为……写……”。如：
He often writes articles for newspapers. 他常为报社写文章。
The book was written specifically for children. 这本书是专门为儿童编写的。
The book is compiled especially for beginners. 这本书是专门为初学者编写的。
He writes a weekly column for the local newspaper. 他为当地报纸每周专栏撰写文章。
2. 所以 Write the letter for each other 的意思是：

请为你们彼此写好这封信。

或：你们相互为对方写好这封信。
3. 注意它与 write sth to sb 的用法不一样。比较：
Please write me the result.=Please write the result to me. 请把结果写给我。
He wrote her some letters.=He wrote some letters to her. 他给她写了几封信。