It can be said that in the poem "Sweeney Among the Nightingales," poet T. S. Eliot shows an appreciation for the fallen character Apeneck Sweeney in his predicament simply because Eliot knows the poem represents an honest portrayal of a weak man in such a predicament.
The poem "Sweeney Among the Nightingales" is about a man named Sweeney being seduced in a brothel by two prostitutes. Though Sweeney tries to escape, he succumbs in the end, making Sweeney an antihero. Eliot depicts Sweeney's fall in the brothel through using allusions to Agamemnon's fall in both Homer's Odyssey and Aeschylus's Agamemnon (eNotes, "Themes and Meanings"). Agamemnon was King of Argos and commander of the Greek army in the Trojan war and recognized as a very powerful and noble warrior. However, when he returned home after the war, his wife Clytemnestra murdered both him and his war prize mistress Cassandra (Encyclopaedia Britannica, "Agamemnon").
We especially see Eliot likening Sweeney's fall to the prostitutes to Agamemnon's fall in the final stanza:
And sand within the bloody wood
When Agamemnon cried aloud
And let their liquid siftings fall
To stain the stiff dishonoured shroud.
We also know Sweeney tried to escape the prostitutes due to the lines "[l]eaves the room and reappears / Outside the window." However, we also know Sweeney did not successfully remain outside of the brothel because the phrase "stiff dishonoured shroud" can be interpreted as having sexual connotations and references to the loss of sexual virtue.
We can particularly see Eliot's appreciation for Sweeney displayed in his comparisons of Sweeney to Agamemnon. Though both were fallen men, they are also heroes and therefore worthy of appreciation.