By a semi-simplicial set I mean a simplicial set without degeneracies. In such a thing we can define horns as usual, and thereby "semi-simplicial Kan complexes" which have a filler for every horn. Unlike when degeneracies are present, we have to include 1-dimensional horns: having fillers for 1-dimensional horns means that every vertex is both the source of some 1-simplex and the target of some 1-simplex.

I have been told that it is possible to choose degeneracies for any semi-simplicial Kan complex to make it into an ordinary simplicial Kan complex. For instance, to obtain a degenerate 1-simplex on a vertex $x$, we first find (by filling a 1-dimensional 1-horn) a 1-simplex $f\colon x\to y$, then we fill a 2-dimensional 2-horn to get a 2-simplex $f g \sim f$, and we can choose $g\colon x\to x$ to be degenerate. But obviously there are many possible choices of such a $g$.

I have three questions:

Where can I find this construction written down?

Is the choice of degeneracies unique in some "up to homotopy" sense? Ideally, there would be a space of choices which is always contractible.

Does a morphism of semi-simplicial Kan complexes necessarily preserve degeneracies in some "up to homotopy" sense? (A sufficiently positive answer to this would imply a corresponding answer to the previous question, by considering the identity morphism.)