Writing recommendation letters for two fellowship candidates, conflict of interest? Let's say there are two applicants to a private foundation fellowship which requires a faculty sponsorship and additional recommendation letters. Would there be a conflict of interest for a faculty to sponsor one applicant with an accompanying letter, and write a recommendation letter for the other applicant?

I thought there would be no conflict, but my fellow faculty member disagrees. What are the possible issues with this situation?
There are no conflicts of interest for the letters. Many of us are at large universities where it is very likely that multiple students will ask for letters for the same fellowship/job. As long as you don't say that multiple students are each "the best student you've had in twenty years," you aren't perjuring yourself. You don't have to select a favorite.

Student sponsorship / mentoring-fellowship is more complex. It's often best to have the student write to the search committee and ask. Some places are fine with faculty sponsoring multiple students, others aren't. Some will note this explicitly in their instructions to faculty. It's best to check with the organization.

So I don't think there will a problem with the faculty writing two letters but only serving as sponsor for one student. And the fact they are not  the sponsor for the other student won't be seen as necessarily negatively prejudicial. Students select  their sponsors for a variety of reasons and vice versa.