  In the theory of coalgebras $C$ over a ring $R$, the rational functor relates
the category of modules over the algebra $C^*$ (with convolution product) with
the category of comodules over $C$. It is based on the pairing of the algebra
$C^*$ with the coalgebra $C$ provided by the evaluation map $\ev:C^*\ot_R C\to
R$. We generalise this situation by defining a {\em pairing} between
endofunctors $T$ and $G$ on any category $\A$ as a map, natural in $a,b\in \A$,
$$\beta_{a,b}:\A(a, G(b)) \to \A(T(a),b),$$ and we call it {\em rational} if
these all are injective. In case $\bT=(T,m_T,e_T)$ is a monad and
$\bG=(G,\delta_G,\ve_G)$ is a comonad on $\A$, additional compatibility
conditions are imposed on a pairing between $\bT$ and $\bG$. If such a pairing
is given and is rational, and $\bT$ has a right adjoint monad $\bT^\di$, we
construct a {\em rational functor} as the functor-part of an idempotent comonad
on the $\bT$-modules $\A_{\rT}$ which generalises the crucial properties of the
rational functor for coalgebras. As a special case we consider pairings on
monoidal categories.
