  This paper constructs cospecialization homomorphisms between the (p')
versions of the tempered fundamental group of the fibers of a smooth morphism
with polystable reduction (the tempered fundamental group is a sort of analog
of the topological fundamental group of complex algebraic varieties in the
p-adic world). We studied the question for families of curves in another paper.
To construct them, we will start by describing the pro-(p') tempered
fundamental group of a smooth and proper variety with polystable reduction in
terms of the reduction endowed with its log structure, thus defining tempered
fundamental groups for log polystable varieties.
