order theory - Prove that, for all ordered sets P, Q and R,

$\langle P \rightarrow \langle Q \rightarrow R\rangle \rangle \cong \langle P \times Q \rightarrow R\rangle $

where $\langle Q \rightarrow R\rangle $ is the set of all order-preserving maps from Q to P; $\cong$ is order-isomorphic symbol; $\times$ 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 $\varphi\in\langle P\to\langle Q\to R\rangle\rangle$; then for each $p\in P$, $\varphi(p)$ is an order-preserving map from $Q$ to $R$. Define

$$\widehat\varphi:P\times Q\to R:\langle p,q\rangle\mapsto\big(\varphi(p)\big)(q)\;.$$

Show that the map $\varphi\mapsto\widehat\varphi$ is the desired isomorphism.