Brilliant explanation of why Bhagavan Shree NaraSimha is Half-a-Lion-and-Half-a-Man from "Shree LakshmiNaraSimha-Panchasat".
"Shree LakshmiNaraSimha-Panchasat", Verse 10:
प्रह्लादे नरतां निरीक्ष्य नरतां तत्याज्ञ नैव स्वयं
तत्ताते मृगतामुदीक्ष्य मृगतां अङ्गीचकारानाने ।
रुपव्याकरणं समस्तजगतां यत्कृत्यमाहुर्बुधाः
तद्रूपं न तृगस्य नौव मनुजस्येत्यद्भुतं दृश्यते । ।
prahlAde naratAM nirIkShya naratAM tatyAj~na naiva svayaM
tattAte mRRigatAmudIkShya mRRigatAM a~NgIchakArAnAne |
rupavyAkaraNaM samastajagatAM yatkRRityamAhurbudhAH
tadrUpaM na tRRigasya nauva manujasyetyadbhutaM dRRishyate | |
Explanation: "It is indeed a matter of wonder that the Lord’s sacred body as Narasimhan does not relate to that of either an animal or a human entirely. We witness this mystery.
The knowledgeable ones say that it is His deliberate act to differentiate between all the objects of His creation and determine which is which and how to relate to them.
PrahlAdan had human qualities (naratAm). Therefore, the Lord did not abandon the attributes of the human being and assumed a human form below his neck (prahlAde naratAm nirIkshya naratAm svayam tatyAja naiva).
PrahlAda’s father exhibited the qualities of an animal (mrgatAm). Hence, the Lord took on a fierce animal face to relate to HiraNyan and to frighten him out of his wits and destroy him (tat tAte mrgatAm udIkshya mrgatAm Anane angIcakAra)".