gs background check discrepancy

I am a CS major and don't really know much about the world of finance, so please forgive me if i say anything dumb or if this entire question is stupid.

I accepted an offer for summer technology analyst at GS and recently got the link to do my background check. The only thing i'm concerned about is my summer "internship" last year. I did some unpaid work for a family friend who was a director at a relatively well known hedge fund, but was in the process of starting his own hedge fund. During the summer, I worked on a tool for his start up hedge fund, which had not officially been incorporated (is this the right word?) yet. When I applied to gs, I listed the company that he had been working at as the company I worked at on my resume, since he did not have the name of the company chosen yet when I sent in my application. Since then, he has incorporated the company (and chosen a name).

When I fill out my information on my background check, I plan to use the name of the company that he has started. Will this be an automatic red flag since it is different from what was listed on my resume? If they do ask me about this, will they accept my explanation? It was not very "official" so there are no official forms or pay stubs or anything, but my family friend will definitely vouch for me if they contact him.

Again, I'm sorry if this is a dumb question or if this is not the right place to ask.