  Quasi-hereditary were introduced by L. Scott \cite{Scott, CPS1,CPS2} in order
to deal highest weight categories as they arise in the representation theory of
semi-simple complex Lie algebras and algebraic groups, and they have been a
very important tool in the study of finite-dimensional algebras. On the other
hand, functor categories were introduced in representation theory by M.
Auslander [A], [AQM] and used in his proof of the first Brauer-Thrall
conjecture [A2] and later on used systematically in his joint work with I.
Reiten on stable equivalence [AR], [AR2] and many other applications. Recently,
functor categories were used in [MVS3] to study the Auslander-Reiten components
of finite-dimensional algebras. The aim of the paper is to introduce the
concept of quasi-hereditary category, and we can think of the components of the
Auslander-Reiten components as quasi-hereditary categories. In this way, we
have applications to the functor category $\mathrm{Mod}(\mathcal{C} )$, with
$\mathcal C$ a component of the Auslander-Reiten quiver.
