  Let $E$ be an Anderson $A$-module over $\mathbb{C}_{\infty}$. The period
lattice of $E$ is related to its module of special functions by means of a
non-canonical isomorphism introduced by the authors in [GM21]. In this paper,
we explain how a modification of the inverse map is canonical by interpreting
it as a residue morphism along the characteristic graph. This phenomenon has
already been observed in various situations. The main innovation of this text
is that of costability (costable admissible opens, costable site, etc.) which
provides a convenient framework to develop the notion of sheaves of
$E(\mathbb{C}_{\infty})$-valued meromorphic functions on the rigid analytic
plane.
