We show that the path construction integration of Lie algebroids by Lie
groupoids is an actual equivalence from the category of integrable Lie
algebroids and complete Lie algebroid comorphisms to the category of source
1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to
construct an actual symplectization functor in Poisson geometry. We include
examples to show that the integrability of comorphisms and Poisson maps may not
hold in the absence of a completeness assumption.Comment: 28 pages, references adde
