Open marcbezem opened 2 years ago
I don't understand all of what you say, but it seems to me that the proof currently addresses uniqueness. And since the basepoint of the middle object is obtained by applying p to the basepoint of X, the identity between the images preserves basepoints.
The question in the first point came up when I tried to understand why $\circ$ is an equivalence. The problem is the third argument $\alpha$. If, for example, the image has a non-trivial auto-equivalence $e$ such that $ep = p$ and $je = j$, then $jep = jp$, and composition would not be an equivalence. (Here I mean $p$ and $j$ as in the image factorization.) So it seems essential that we prove in Theorem 3.10.17 that $e$ is unique. We do this by cancelling the $n$-connected map $p$ on the right: $ep = p$ implies $e = id$.
The title refers to \cref{con:im}, currently Construction 5.3.11.