falcon vs pigeon hawk

falcon and pigeon hawk both are nouns.

Word	Noun	Adjective	Verb	Adverb
falcon	Yes	No	Yes	No
pigeon hawk	Yes	No	No	No

As nouns, pigeon hawk is a hyponym of falcon; that is, pigeon hawk is a word with a more specific, narrower meaning than falcon:

falcon: diurnal birds of prey having long pointed powerful wings adapted for swift flight
pigeon hawk: small falcon of Europe and America having dark plumage with black-barred tail; used in falconry

falcon (noun)	pigeon hawk (noun)
diurnal birds of prey having long pointed powerful wings adapted for swift flight	small falcon of Europe and America having dark plumage with black-barred tail; used in falconry

falcon (verb)	pigeon hawk (verb)
hunt with falcons