  We introduce the notion of pre-weight structure on a triangulated category
and study the corresponding pseudo-identities. We propose the notion of
canonical derived equivalence between algebras that are not necessarily flat,
which is associated to a tilting complex. In the flat situation, canonical
derived equivalences coincide with standard derived equivalences in the sense
of Rickard. We prove that any derived equivalence starting from a hereditary
algebra is canonical. The key tool is a general factorization theorem: any
derived equivalence is uniquely factorized as a pseudo-identity followed by a
canonical derived equivalence.
