isProper f
A morphism of varieties is proper if it is universally closed. For a toric map $f : X \to Y$ corresponding to the map $g : N_X \to N_Y$ of lattices, this is equivalent to the preimage of the support of the target fan under $g$ being equal to the support of the source fan. For more information about this equivalence, see Theorem 3.4.11 in Cox-Little-Schenck's Toric Varieties.
We illustrate this method on the projection from the second Hirzebruch surface to the projective line.
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The second example shows that the projection from the blow-up of the origin in the affine plane to affine plane is proper.
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The natural inclusion of the affine plane into the projective plane is not proper.
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To avoid repeating a computation, the package caches the result in the CacheTable of the toric map.