  In this paper, we give a simple criterion to verify that functions of the
form $e^g$ are in the Hayman class when $g$ is a power series with nonnegative
coefficients. Thus, using the Hayman and B\'aez-Duarte formulas, we obtain
asymptotics for the coefficients of generating functions that arise in many
examples of set construction in analytic combinatorics. This new criterion
greatly simplifies that obtained previously by the authors.
