⟨P→⟨Q→R⟩⟩≅⟨P×Q→R⟩
where ⟨Q→R⟩ is the set of all order-preserving maps from Q to P; ≅ is order-isomorphic symbol; × is cartesian product symbol.
Besides I am not good at building bijection to prove isomorphism, I hope you can teach me some technique on this issue.
Answer
Suppose that φ∈⟨P→⟨Q→R⟩⟩; then for each p∈P, φ(p) is an order-preserving map from Q to R. Define
ˆφ:P×Q→R:⟨p,q⟩↦(φ(p))(q).
Show that the map φ↦ˆφ is the desired isomorphism.
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