  Poincar\'e maps and suspension flows are examples of fundamental
constructions in the study of dynamical systems. This study aimed to show that
these constructions define an adjoint pair of functors if categories of
dynamical systems are suitably set. First, we consider the construction of
Poincar\'e maps in the category of flows on topological manifolds, which are
not necessarily smooth. We show that well-known results can be generalized and
the construction of Poincar\'e maps is functorial, if a category of flows with
global Poincar\'e sections is adequately defined. Next, we consider the
construction of suspension flows and its functoriality. Finally, we consider
the adjointness of the constructions of Poincar\'e maps and suspension flows.
By considering the naturality, we can conclude that the concepts of topological
equivalence or topological conjugacy of flows are not sufficient to describe
the correspondence between map dynamical systems and flows with global
Poincar\'e sections. We define another category of flows with global Poincar\'e
sections and show that the suspension functor and the Poincar\'e map functor
form an adjoint equivalence if these categories are considered. Hence, a
categorical correspondence between map dynamical systems and flows with global
Poincar\'e sections is obtained. This will enable us to better understand the
connection between map dynamical systems and flows.
