Definition
A morphism in the category of Poisson manifolds is a Poisson map, which is a map between Poisson manifolds that respects the Poisson brackets.
That is
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A morphism in the category of Poisson manifolds is a Poisson map, which is a map Φ:M1→M2 between Poisson manifolds that respects the Poisson brackets.
That is
{f∘Φ,f∘Φ}1={f,g}2∘Φ∀f,g∈C∞(M2).