  In the setting of a Drinfeld module $\phi$ over a curve $X/\mathbb{F}_q$ with
an $\mathbb{F}_q$-rational point $\infty$, we use a functorial point of view to
define \textit{Drinfeld eigenvectors}, a generalization of the so called
"special functions" introduced by Angl\`es, Ngo Dac and Tavares Ribeiro. We
adopt an analogous approach with the dual Drinfeld module $\phi^*$ to define
\textit{dual Drinfeld eigenvectors}, and we prove that both functors are
representable with universal objects $\omega_\phi$ and $\zeta_\phi$,
respectively. We prove that $\zeta_\phi$ can be expressed as an Eisenstein-like
series over the period lattice; moreover, for all integers $i$ we define dot
products $\zeta_\phi\cdot\omega_\phi^{(i)}$, which are meromorphic differential
forms over a base change of the curve $X\setminus\infty$, and prove they are
actually rational. These two results complete the investigation started by the
author in a previous work on Drinfeld modules of rank $1$, albeit with very
different methods. In this paper, although we are not able to compute the
divisors of those differential forms for arbitrary curves, we give a working
description of the generating series
$\sum_{i\in\mathbb{Z}}\zeta_\phi\cdot\omega_\phi^{(i)}\tau^i$, and we apply it
to compute the forms $\zeta_\phi\cdot\omega_\phi^{(i)}$ in the case of genus
$0$.
  We also describe the module of special functions for Anderson modules, as
already done by Gazda and Maurischat, and prove their conjecture about the
invertibility of special functions for Drinfeld modules of rank $1$.
