  We consider identification and inference about a counterfactual outcome mean
when there is unmeasured confounding using tools from proximal causal inference
(Miao et al. [2018], Tchetgen Tchetgen et al. [2020]). Proximal causal
inference requires existence of solutions to at least one of two integral
equations. We motivate the existence of solutions to the integral equations
from proximal causal inference by demonstrating that, assuming the existence of
a solution to one of the integral equations, $\sqrt{n}$-estimability of a
linear functional (such as its mean) of that solution requires the existence of
a solution to the other integral equation. Solutions to the integral equations
may not be unique, which complicates estimation and inference. We construct a
consistent estimator for the solution set for one of the integral equations and
then adapt the theory of extremum estimators to find from the estimated set a
consistent estimator for a uniquely defined solution. A debiased estimator for
the counterfactual mean is shown to be root-$n$ consistent, regular, and
asymptotically semiparametrically locally efficient under additional regularity
conditions.
