We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the of integrable Lie algebroids and complete Lie algebroid comorphisms to the 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.
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Alberto S. Cattaneo, Benoit Dherin, Alan Weinstein, Integration of Lie algebroid comorphisms. Port. Math. 70 (2013), no. 2, pp. 113–144DOI 10.4171/PM/1928