  There are different categorical approaches to variations of transition
systems and their bisimulations. One is coalgebra for a functor G, where a
bisimulation is defined as a span of G-coalgebra homomorphism. Another one is
in terms of path categories and open morphisms, where a bisimulation is defined
as a span of open morphisms. This similarity is no coincidence: given a functor
G, fulfilling certain conditions, we derive a path-category for pointed
G-coalgebras and lax homomorphisms, such that the open morphisms turn out to be
precisely the G-coalgebra homomorphisms. The above construction provides
path-categories and trace semantics for free for different flavours of
transition systems: (1) non-deterministic tree automata (2) regular
nondeterministic nominal automata (RNNA), an expressive automata notion living
in nominal sets (3) multisorted transition systems. This last instance relates
to Lasota's construction, which is in the converse direction.
