Admissible layers Q-construction

Question

I have a question which seems to be extremely trivial but for some reason I don't get it and am very confused about it. In the paper by Quillen, "Higher Algebraic K-theory I" page 94 top. Quillen claims that if we have the Q-construction $QM$ for some exact category $M$ and an arbitrary object $P$. The over category $QM/P$ is equivalent to the ordered set of admissible layers $(M_0,M_1)\leq (M_0',M_1')$ iff $M_0'\leq M_0 \leq M_1 \leq M_1'$. So it is basically a poset. But how can this be true, when every object in the over category has automorphism. Let's look at $P'\mapsto P$ which is given by an admissible monomorphism. This element has automorphisms given by the automorphisms of $P'$, just pre-compose it. But in that poset nothing has any automorphisms? So I don't get why we suddenly forget automorphisms? is it supposed to be a homotopy equivalence? If so why?

Answer 1

Let $i: P' \to P$ be the inclusion. Regard $i$ as an object in the comma category $QM/P$. An automorphism $g$ of $P'$ gives rise to an automorphism of $i$ in $QM/P$ only if $i \circ g = i$ in $M$, and that happens only if $g = 1$.