  We prove that to every inclusion $A\hookrightarrow L$ of Lie algebroids over
the same base manifold $M$ corresponds a Kapranov dg-manifold structure on
$A[1]\oplus L/A$, which is canonical up to isomorphism. As a consequence,
$\Gamma(\Lambda^\bullet A^\vee\otimes L/A)$ carries a canonical $L_\infty[1]$
algebra structure whose unary bracket is the Chevalley--Eilenberg differential
corresponding to the Bott representation of $A$ on $L/A$ and whose binary
bracket is a cocycle representative of the Atiyah class of the Lie pair
$(L,A)$. To this end, we construct explicit isomorphisms of
$C^\infty(M)$-coalgebras
$\Gamma\big(S(L/A)\big)\xrightarrow{\sim}\frac{\mathcal{U}(L)}{\mathcal{U}(L)\Gamma(A)}$,
which we elect to call Poincar\'e--Birkhoff--Witt maps. These maps admit a
recursive characterization that allows for explicit computations. They
generalize both the classical symmetrization map
$S(\mathfrak{g})\to\mathcal{U}(\mathfrak{g})$ of Lie theory and (the inverse
of) the complete symbol map for differential operators. Finally, we prove that
the Kapranov dg-manifold $A[1]\oplus L/A$ is linearizable if and only if the
Atiyah class of the Lie pair $(L,A)$ vanishes.
