Vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair

We investigate vertical isomorphisms of Fedosov dg manifolds associated with a Lie pair $(L,A)$, i.e. a pair of a Lie algebroid $L$ and a Lie subalgebroid $A$ of $L$. The construction of Fedosov dg manifolds involves a choice of a splitting and a connection. We prove that, given any two choices of a splitting and a connection, there exists a unique vertical isomorphism, determined by an iteration formula, between the two associated Fedosov dg manifolds. As an application, we provide an explicit formula for the map ${\mathrm{p}\mathrm{b}\mathrm{w}}_2^{-1}\circ {\mathrm{p}\mathrm{b}\mathrm{w}}_1$ associated with two Poincaré–Birkhoff–Witt isomorphisms that arise from two choices of a splitting and a connection.