On Rational Pairings of Functors

In the theory of coalgebras $C$ over a ring $R$, the rational functor relates the category of modules over the algebra $C^{\ast }$ (with convolution product) with the category of comodules over $C$. It is based on the pairing of the algebra $C^{\ast }$ 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 $\text{A}$ as a map, natural in $a,b\in \text{A}$, $${\beta }_{a,b}:\text{A}(a,G(b))\to \text{A}(T(a),b),$$ and we call it {\em rational} if these all are injective. In case $\text{bT}=(T,m_T,e_T)$ is a monad and $\text{bG}=(G,{\delta }_G,{\text{ve}}_G)$ is a comonad on $\text{A}$, additional compatibility conditions are imposed on a pairing between $\text{bT}$ and $\text{bG}$. If such a pairing is given and is rational, and $\text{bT}$ has a right adjoint monad ${\text{bT}}^{\text{di}}$, we construct a {\em rational functor} as the functor-part of an idempotent comonad on the $\text{bT}$-modules ${\text{A}}_{\text{rT}}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.