  We study the fixed point problem for a system of multivariate operators that
are coordinate-wise monotone (i.e., nondecreasing or nonincreasing in each of
the variables, independently), in the setting of quasi-ordered sets. We show
that this problem is equivalent to the fixed point problem for a mixed monotone
operator that can be explicitly constructed. As a consequence, we obtain a
criterion for the existence and uniqueness of solution to our problem, in the
setting of partially ordered metric spaces. To validate our results, we provide
an application to a first-order differential system with periodic boundary
value conditions. A direct consequence that follows from our paper is that all
the separate recent developments on the subject of tripled, quadrupled or
multidimensional fixed points are but particular aspects of a single, unified
and much simpler approach.
