Let $D\_2$ be the topological operad of little disks. This operad can be modelled "combinatorially" in terms of an operad of groupoids called $\newcommand{\PaB}{\mathbf{PaB}}\PaB$, the operad of *parenthesized braids*. An object of the groupoid $\PaB(n)$ is a complete parenthesization of a permutation of the symbols $\{1,...,n\}$, e.g.
$$ ((24)(13)5) \in \mathrm{ob} \,\PaB(5),$$
and morphisms are $n$-strand braids, such that the start- and endpoints of each strand are labeled by the same element of $\{1,...,n\}$. Composition in this operad is given by "cabling". The connection to the little disk operad is that one can quite easily write down a morphism of operads 
$$ \PaB \to \Pi\_1(D\_2),$$
where $\Pi\_1(-)$ denotes the fundamental groupoid, such that $\PaB(n) \to \Pi\_1(D\_2)(n)$ is an equivalence of categories for all $n$.


If we denote by $N$ the composition of the nerve functor with the geometric realization, then $N\PaB$ is a topological operad.


*Fact: $N\PaB$ is equivalent to $D\_2$, i.e. there is a zig-zag of operad morphisms between $N\PaB$ and $D\_2$, each of which is arity-wise a weak equivalence.*


In the literature this fact is in several places proven by appealing to Fiedorowicz's recognition principle, see e.g. Dmitry Tamarkin, Formality of chain operad of little discs, Section 2. I don't understand why and I guess I am missing something: it seems to me that there is a much more direct argument. Indeed the map $\PaB \to \Pi\_1(D\_2)$ gives a homotopy equivalence 
$$N\PaB \to N\Pi\_1(D\_2),$$
moreover, the unit of the adjunction between $\Pi\_1$ and $N$ gives a morphism 
$$D\_2 \to N\Pi\_1(D\_2)$$
which is also an equivalence since $X \to N\Pi\_1(X)$ is a homotopy equivalence for any $K(\pi,1)$-space, such as $D\_2(n)$. What am I missing?



Identifying the little disk operad with parenthesized braids
It seems to me that the operad morphism from **PaB** to $\Pi\_1(D\_2)$ works in positive arity but does not respect composition with the $0$-ary operation on the object level.


