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?

Homotopy of Operads and Grothendieck-Teichmüller Groupsby Fresse, more precisely the statement is given in Proposition 6.2.2 ($\mathtt{PaB}$ is isomorphic to a suboperad of $\pi \mathtt{D}_2$). The unitary case (with nonempty arity zero) is also dealt with. $\endgroup$