  Some aspects of basic category theory are developed in a finitely complete
category $\C$, endowed with two factorization systems which determine the same
discrete objects and are linked by a simple reciprocal stability law. Resting
on this axiomatization of final and initial functors and discrete
(op)fibrations, concepts such as components, slices and coslices, colimits and
limits, left and right adjunctible maps, dense maps and arrow intervals, can be
naturally defined in $\C$, and several classical properties concerning them can
be effectively proved.
  For any object $X$ of $\C$, by restricting $\C/X$ to the slices or to the
coslices of $X$, two dual "underlying categories" are obtained. These can be
enriched over internal sets (discrete objects) of $\C$: internal hom-sets are
given by the components of the pullback of the corresponding slice and coslice
of $X$. The construction extends to give functors $\C\to\Cat$, which preserve
(or reverse) slices and adjunctible maps and which can be enriched over
internal sets too.

Category Theory