Consider a position that holds that there are real moral objects grasped by a form of moral perception or intuition. A widespread fear about such a position is that it demands of us something "spooky". That's a rather silly fear in itself, but it is in any case unfounded. This is true even where the perception of moral objects is seen as fairly closely analogous to the perception of sensible objects.
We can think of perception of sensible objects, roughly, but for our purposes adequately, in quasi-Kantian terms. I can think about the external world as an external world because I seem to perceive certain things, and I have a notion of physical object in light of which I can interpret my perceptions as presenting something objectively to me. To get knowledge about these objective physical somethings, then, I simply need to be able to fill out this general notion of physical object with a more specific concept that makes clear how I can come to know them; namely, a concept of body capable of being investigated scientifically. This in hand, I have the external world, understood as an external world; it is objective and I can, due to scientific investigation, come to have knowledge of it through my perceptions. Sensible intuition in this light becomes a means of knowledge, a way of grasping truths, about objective things.
A more controversial example may clarify what this could indicate about a certain view of moral intuition. It is tempting, and has always been tempting, to hold that we have some sort of special direct, particularly mathematical intuition by means of which we come to mathematical knowledge. This intelligible intuition might seem at first an odd thing. But it's entirely possible to give a decent account of what such an intelligible intuition would be, one that follows fairly closely the analogy of sensible intuition: Just as there is a notion of physical object making possible the interpretation of some phenomena involved in (as it were) our being forced to recognize some things as true, so there is a notion of mathematical object making possible the interpretation of some phenomena involved in (as it were) our being forced to recognize some things as true. We can call this a broadly Gödelian account of mathematical intuition. There are several different ways we might think of filling out this notion of mathematical object; an account in which it is filled out with the concept of the iterative set can be called a properly Gödelian account of mathematical intuition, because it is, in fact, one way of stating Gödel's own view of the matter.
It is important to understand what is, and what is not, given to us by any sort of broadly Gödelian account of mathematical intuition, including Gödel's own. Such mathematical inuition does not give us direct access to mathematical truths. Mathematical intuition cannot give us truth, only objectivity, and the two are not the same. Think of sensible intuition again. When we perceive a sensible object, we are not thereby given direct access to truth about the object itself; sensible intuition does not give us direct insight into the nature of the object. Rather, it gives us an object, something objective into which we can gain insight. We don't get knowledge directly from it; we get something we can come to have knowledge about through it. And the analogy holds here. Just as we don't think sensible intuition gives us direct access to physical truths without inquiry, so we don't have direct access to mathematical truths without inquiry. This is why Gödel always regards the analogy to imply that, just as we gain knowledge of the physical world by testing hypotheses about physical objects, so we must gain knowledge of the mathematical world by testing hypotheses about mathematical objects. He is right to think that the properly Gödelian account of mathematical intuition implies this, because all that mathematical intuition can directly give us on that account is an object for inquiry.
Now, to say that is slightly misleading, because mathematical intuition can on this account give us something more; but what it gives is derivative of objectivity, and certainly falls short of truth. This is a standard of plausibility. When I perceive a physical object, I don't merely get an object qua object presented to me; I also get a way it seems to be. This allows me to make plausible presumptions, guesses, and hypotheses about it, and to say that other presumptions, guesses, and hypotheses are implausible. This plausibility is not truth; our recognizing a claim as plausible is not knowledge. To get to truth we need the claim not merely to be something that can be plausibly made about the object; it has to be genuinely adequate to the object. But it is through inquiry and investigation that the plausible becomes the adequate. Indeed, one could argue that this plausibility is necessary for any knowledge of the objects themselves; without it, our testing would be merely a testing of whether a claim is consistent with the objects. It is because the appearances of the objects constrain what's plausible to say about them that we can say that our tested conclusions are not merely consistent with the objects, abstractly considered, but linked to them through the way the objects seem to be. In other words, we don't merely find accounts consistent with the phenomena, we find accounts consistent with the phenomena in more natural and less natural ways. Further inquiry can winnow them further. Thus sensory perception does not merely give us objectivity; it gives us a guide to inquiry by making more or less plausible claims that can be further tested. Principles discovered scientifically begin to force themselves on us as something we have to recognize as true, at least to a degree of approximation through the combination of (on the one hand) the objectivity and constraint on plausibility we find in sensible intuition and (on the other) the discoveries we make while testing out in various ways what's plausible in light of intuition. So it is, one could say, with mathematical intuition.
This is perfectly generalizable, I believe, to other cases in which appeals to intuition are made as if they somehow adjudicate among claims. (I don't commit to the quasi-Kantian way of stating the claim as being the best way to state it; I think, in fact that there are ways that allow us to do it more precisely. The quasi-Kantian way, however, is convenient in a number of ways, and, as I said, seems sufficient for our purpose here, which is just to sketch out why one can hold the original claim in a perfectly reasonable way.) To do so what such intuition gives us would have to be objective and make some things plausible and others not; and if it can be argued about, it would appear there is a way of testing them. And that's all the pieces required. So it would appear that any appeal to moral intuitions as somehow providing evidence or guidelines in philosophical disputes about moral matters would have to exhibit all the elements needed for the claim made at the beginning of this post. Moreover, I think it's clear that this is a very needful thing. When it comes down to argument, a great deal depends on what concept one uses to cash out the notion of the particular type of object in question. Someone with a properly Gödelian account will not be engaging in the same type of inquiry as someone with some other concept of mathematical object than the iterative set, even if that other person also holds a broadly Gödelian account. Moreover, some concepts will be better than others, as being more fruitful in their consequences (again, through testing). It may be that all appeals to moral intuition take a fairly unified object; or (as I think is the case) we will find that appeals to moral intuition actually break down into several different things. And we would have to see whether in so breaking down they break down into anything with an object well-defined enough to allow a close analogy to the above cases. Again, I think this would prove to be so; but any appeal to moral intuition needs to give some sort of account establishing the features previously noted. If it does, it's entirely reasonable; the only question will be whether there is a better account. And only on some such account can appeal to moral intuitions carry any genuine force. (As I said, this is perfectly generalizable to any appeal to intuitions.)
Of course, some people worry not about the objects but about what's involved in this intuiting, worrying that this is to propose some sort of "spiritual intuition," to use Hume's phrase. That's an interesting question, but it's putting the cart before the horse to take it as a problem for the claim rather than a question for further research. It's pointless to worry about what would be involved in such intuiting until we've established something about it. Then we can see what the evidence tells us about it, in that light. Before then, we don't really have an objection, just a prejudice.