What exactly are the challenges to morality Nietzsche presents in the Genealogy of Morality? Is there a specific philosopher Nietzsche is responding to (like Kant, Aristotle, Plato, or Hobbes)? And is it fair to interpret the "morality" Nietzsche speaks about as the same "morality" these other philosophers generally refer to?
Nietzsche often does not quote and or refer to his sources. At best you can say that he expects to pick up the references, and at worst you can say he is simply purloining the ideas of others without due credit and burnishing his own flame at the expense of others.
One might say, for a man interested in genealogy he's singularly uninterested in the genealogy of his own; perhaps, he sees himself sui generis? That would figure given his eulogy of ubermenschen - and did he count himself amongst them? Maybe, maybe, maybe - we don't know! But he certainly made himself out to be something uber uniqo and uber fantastico; that Greek spirit certainly went his head but whereas most sober up, one wonders whether he ever did...
For example, here's something that I've just read from a footnote in an essay by Ortega y Gasset:
In passing, let us seize the opportunity to see, from the altitude of this vision, the element of frivolity, and even of vulgarity, in N's famous imperative: 'Live dangerously'. Which furthermore, is not N's but the exaggeration of an old Italian Renaissance saying, Aretinos famous motto: Vivere risolutamente. Because N does not say 'Live on the alert', which would have been good, but 'Live dangerously.' And this shows that despite his genius, he did not know that the very substance of our life is danger and hence it is rather affected, and supererogatory, to propose as something new, added, and original that we should seek and collect danger. An idea, furthermore, which is typical of that period called 'fin de siècle'
And mind you, Ortega happened to be a fan of Nietzsche! That didn't stop him from disapproving of his philosophy of excess, of turning words upside down, of giving words wings, and of borrowing the words of others - unacknowledged. I'd have whispered in his ear - Nietschze, do you remember the story of Icarus? But would a man like him take any notice? One very much doubts it.
I also recall reading somewhere that an Irish playwright and politician - unfortunately I don't recall the name right now - in the 17th C was critical of Machivelli describing Christianity as a religion of slaves. This of course is interesting in the context of Nietzsches philosophy as he famous for saying this (and then his followers and disciples - they don't seem often to get far past this point for some reason). The playwrights riposte, and maybe the politician in him too, was that Christianity made it that much more difficult for tyrants to rule; that they so often do, shows the strength of tyranny...
Martin Luthor King said this philosophy was something he had to struggle through when he was doing his thesis in the seminary. That critique obviously hit home - one might say as it was meant to. Coming from a world only recently emancipated from slavery but still living like second class or third class citizens - it would, wouldn't it? I mean not only did he come from a family of slaves, and a race of slaves, and that he was the colour of a slave but he was metaphysically a slave! Not just a slave legally, or in fact, but slave all the way through to his soul. No wonder, it jolted him. I don't recall what he thought in response to it except that he was able to go past it. Given your interest in the challenges that N puts to traditional morality of one kind or another it might be worth looking to see what he had to say about his own struggles with this.
What is Nietzsche's view of morality? As the name suggests, Nietzsche's Genealogy of Morality is concerned with looking at the historical lineage of different moral theories. Nietzsche holds the view that humans are affected by powerful inbuilt drives, and the most basic drive is the "will to power". Nietzsche believes that moral codes evolve as conscious normative manifestations of the needs and interests of a particular type of person, and so groups of people tend to formulate moral codes that characterise their own traits as "good" and the traits that they lack as "evil". He argues that people form these moral codes as a weapon to advance their will to power. He identifies two main types of people (the strong and the weak) and two corresponding moral codes (master-morality and slave-morality).
Nietzsche is sceptical of the idea of objective morality, and he sees morality as being an expression of codified will-to-power among people with different characteristics. At one point, he expresses this view of morality with an analogy of the "eagles and lambs":
There is nothing strange about the fact that lambs bear a grudge towards large birds of prey: but that is no reason to blame the large birds of prey for carrying off the little lambs. And if the lambs say to each other, ‘These birds of prey are evil; and whoever is least like a bird of prey and most like its opposite, a lamb, – is good, isn’t he?’, then there is no reason to raise objections to this setting-up of an ideal beyond the fact that the birds of prey will view it somewhat derisively, and will perhaps say: ‘We don’t bear any grudge at all towards these good lambs, in fact we love them, nothing is tastier than a tender lamb.’ (Genealogy of Morality, pp. 25-26)
What moral philosophy is Nietzsche criticising? In Geneaology, Nietzsche heavily criticises the Judeo-Christian ethic of altruism, which he refers to as "slave morality". He argues that this morality evolved as a means to allow the weak to denigrate the strong, and revere their own weakness. He identifies this ethic as originating from the enslaved Jewish people in Egypt, as a weapon for them to undermine their enslavers, but he believe that it has since spread much more broadly and has come to dominate society. Nietzsche is highly critical of this moral code and he clearly regards it with extreme disgust. He argues that the slave morality arises from ressentiment, where a weak/inferior person projects their own inferiority and the source of their pain onto the strong. He refers with contempt to the "priestly caste" that spreads this moral code:
As we know, priests make the most evil enemies – but why? Because they are the most powerless. Out of this powerlessness, their hate swells into something huge and uncanny to a most intellectual and poisonous level. The greatest haters in world history, and the most intelligent, have always been priests: – nobody else’s intelligence stands a chance against the intelligence of priestly revenge.
It was the Jews who, rejecting the aristocratic value equation (good = noble = powerful = beautiful = happy = blessed) ventured, with awe-inspiring consistency, to bring about a reversal and held it in the teeth of the most unfathomable hatred (the hatred of the powerless), saying: ‘Only those who suffer are good, only the poor, the powerless, the lowly are good; the suffering, the deprived, the sick, the ugly, are the only pious people, the only ones saved, salvation is for them alone, whereas you rich, the noble and powerful, you are eternally wicked, cruel, lustful, insatiate, godless, you will also be eternally wretched, cursed and damned!’ ...
We know who became heir to this Jewish revaluation ... With regard to the huge and incalculably disastrous initiative taken by the Jews with this most fundamental of all declarations of war, I recall the words I wrote on another occasion ... – namely, that the slaves’ revolt in morality begins with the Jews: a revolt which has two thousand years of history behind it and which has only been lost sight of because – it was victorious ... (Genealogy of Morality, p. 17)