Well, the thing in itself is that with which our representations correlate, see CPR B45 (bolded mine):
[...] [O]bjects in themselves are not known to us at all, and that what we call outer objects are nothing other than mere representations of our sensibility, whose form is space, but whose true correlate, i.e., the thing in itself, is not and cannot be cognized through them, but is also never asked after in experience.
This sounds awfully like if the transcendental object, that which causes the manifold of intuition - since every representation must be the representation of something - was the thing in itself.
But since the categories (including causation) only apply to objects of experience (ie. representations) and the relations between them, and the thing in itself by definition cannot be part of experience, how should it be able to cause our sensual input? It seems, and that is the main thrust of Schulze's criticism, as if Kant would introduce something as necessary for the coming into being of our representations (and thus a transcendental object), but at the same time forbid that it can cause anything because it is in the wrong category. But you cannot have the cake and eat it, right?
And indeed, Kant was well-aware of that problem. He writes in B309 (bolded mine):
But if, on the contrary, I leave out all intuition, then there still remains the form of thinking, i.e., the way of determining an object for the manifold of a possible intuition. Hence to this extent the categories extend further than sensible intuition, since they think objects in general without seeing to the particular manner (of sensibility) in which they might be given. But they do not thereby determine a greater sphere of objects, since one cannot assume that such objects can be given without presupposing that another kind of intuition than the sensible kind is possible, which, however, we are by no
means justified in doing.
In other words: Since the thing in itself is a mere abstract object of thought (and hence not empirical and, strictly speaking, not even an object) but at the same time we have to think of it as an object "like (our empirical) things" but perceived without the limitations of the particularities of our sensibility, we kind of overstep the allowed application of categories in this case. We cannot help ourselves but doing so, as it were.
Now, does this mean that there is an inner contradiction in Kant? Not necessarily so. He is descriptive here, telling us how our manner of thinking forces us to assume that our representations have a correlating transcendental object and at the same time think that at least structurally, it has to be like our objects of experience, which are our representations.
Therefore, we cannot help but apply the categories because that's how we think, even though we are, strictly speaking, not allowed to do so. He also insists that strictly speaking, it is wrong to think that way. He does write about that nevertheless because he wants to outline the boundaries of possible knowledge, so limiting concepts - concepts which draw the line from outside of possible knowledge - are a common means of Kant. And this does not mean that he claims metaphyical truths about the thing in itself here. As he expresses it in his notes (posthumously published):
If the borderline of transcendental philosophy would be crossed, the assumed principle became transcendent; i.e. the object becomes a no-thing [ein Unding], the concept [Begriff] of it contradicts itself: Because it crosses the borderline of all knowledge: the uttered word is without meaning. (Ak. 21:75)
