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Obtaining the poset of monotone functions

Let $P$ and $Q$ be finite posets. Is there an easy way to obtain the poset (with the natural order) of monotone function from P to Q? Is it possible to obtain also the poset of injective (or surjective) monotone functions?

Special cases would also be interesting such as when P and Q are lattices or total orders.

Obtaining the poset of monotone functions

Let $P$ and $Q$ be finite posets. Is there an easy way to obtain the poset (with the natural order) of monotone function from P to Q? Q via Sage? Is it possible to obtain also the poset of injective (or surjective) monotone functions?

Special cases would also be interesting such as when P and Q are lattices or total orders. orders.