I'm struggling to understand Kant's schematism. Kant says that imagination produces the synthesis of schemata and that schemata are how we can relate intuitions to concepts. He goes on to give the example of the dog and this makes things clearer, however, how is the synthesis of schemata possible concretely: every time we have intuitions, how does imagination perform the synthesis of the exact application of a rule to match our intuition ? As a concrete example: given that we have the intuition of an acute triangle, how is imagination doing the synthesis from the rule (ie. of how to build a triangle) to the exact acute triangle ? Clearly, it is not synthesizing all possible applications of the rule, correct ?
A good point-of-entry into the level of analysis you're after is Kant's situation of intuition in relation to other "similar" terms:
We are in no want of words to denominate adequately every mode of representation, without the necessity of encroaching upon terms which are proper to others. The following is a graduated list of them. The genus is representation in general (representatio). Under it stands representation with consciousness (perceptio). A perception which relates solely to the subject as a modification of its state, is a sensation (sensatio), an objective perception is a cognition (cognitio). A cognition is either an intuition or a conception (intuitus vel conceptus). The former has an immediate relation to the object and is singular and individual; the latter has but a mediate relation, by means of a characteristic mark which may be common to several things. A conception is either empirical or pure. A pure conception, in so far as it has its origin in the understanding alone, and is not the conception of a pure sensuous image, is called notio. A conception formed from notions, which transcends the possibility of experience, is an idea, or a conception of reason.
Earlier, in the Transcendental Aesthetic (IIRC), he says something about space being an intuition first and then a concept on account of its singularity as a total reality, i.e. space is the particular way in which we represent different particulars. (Hence he later in the first Critique floats the abstract possibility of concepts akin to those of space and time, in the sense that he allows for the abstract possibility of numerically differentiating objects in terms other than those of spatial difference, or for numerically identifying objects in terms other than those of identity over time.)
So the first thing to consider is the "similarity" between an imagined representation and an intuition. Both are more particular than general; if you are imagining a triangle, you will not be merely levitating the general definition of a triangle inside your thoughts, but will be actively projecting a specific subtype of triangle onto your imaginal space. Now the obvious gap between imagining and perceiving forestalls imagination as genuinely intuitive as to its arbitrary content, but not as to its pure form. The general triangulation function, and its more specific outputs in imagination together with understanding, are examples of the form of space. Check out the part of the Transcendental Doctrine of Method, the tail end of the first Critique, where Kant talks about the "construction of concepts" that mathematicians engage in.
Note that it is in relation to the faculty of imagination that Kant brings up the mysterian quality of schematism. While Kant does use words in creative(!) ways, still, I think he was trying to be faithful to a lot of the tradition nevertheless, so we might try out reflecting on why Kant chose to use schematics-talk to describe this moment in transcendental logic. I don't know what schema were defined as back then, but nowadays a definition might be offered along the lines of variable theory: a sentence is schematic iff it is composed of more/only unbound variables instead of bound ones, say. (I make no claim that such a definition is necessary or sufficient in order to stating the content of the concept of schematism. I only offer that definition as a highly simplified introduction to the possibility of a better definition.) Modulo Kant, such a description of schematism therefore suggests distinguishing between general and particular variables, with the schematism of transcendental logic having to do with particular variables, the caveat being that these are otherwise the most general particulars(!) of all, viz. the intuitively pure "objects" of space and time. (As you can see, in Kant, generality and particularity are not quite absolute opposites but form an endless sequence of gradation, with absolute particularity being something of a logical ideal to be attained by the endless precision of further and further representation of a numerically individuated/differentiated thing.)
In this sense, there is the general problem of triangles, so to speak, and then the different subtypes (equilateral, isosceles, curved, etc.) are solutions to the "variable" in the "equation" of the "problem."†
EDIT How do we imagine triangles at all? It is customarily remarked that we never imagine a triangle in general, but always one that is at least equilateral or isosceles or what. In Kant, something else is going on, though. Enamored as he was with Newtonian physics, he was also as devout an "infinitesimal analyst" as a philosopher could be in those days.
So, we start with the coordinate system in graphs of functions. The scheme of a circle then becomes x2 + y2 = 1, for example, and the scheme of a triangle is some comparable matter. Now, then, we're going to address the exact issue in terms of standard real analysis for the time being, but keep in mind that Kant was, again, using infinitesimal analysis in the "background," and that use introduces another kind of refinement into the set-up of the problem.
So next, suppose a triangle one of whose (x, y) coordinates is (π, e), another is (e, π), and then another is (π, √2). Depending on how far one is "zoomed in" to the graph of the triangle, one could intuitively detect variations on these coordinates. But suppose "small" differences, like in the trillionth decimal places of any of the numbers. These differences could be either uniform for all the numbers, so that we were speaking of a same-sized triangle just slightly shifted in position vs. the grid; or different among them, so that we would have very slightly different-sized triangles.
Now, a real number is a number that has (or "can" have) at least ℵ0-many digits in its decimal expansion. (I say "at least" because there are normal version of set theory in which the powersets of the zeroth and first alephs are equivalent (indeed, there are otherwise normal set theories in which the powersets of arbitrarily many of the ℵn are equivalent, and many other things besides...).) So "around" our initial triangle, we have Continuum-many slightly differently placed or configured triangles.
So now for Kant, there is an impossible form of infinite synthesis (the one that we would need to be capable of to "objectively" solve the antinomies and the ideal of reason), but he is not necessarily ruling out infinite synthesis in another sense, since(!) he actually offers two definitions of infinity, as that which is not to be added to, or that to which the addition of things never ends. In other words, modulo what Kant says about absolutes, it is absolute infinity that can never be "synthesized" in human epistemic time. But otherwise, human consciousness is able to synthesize continuous sequences, for example, and indeed he would have made something even more peculiarly different of the locations and sizes of the triangles in the cloud: that the variation was not in decimal places of real numbers, but directly in infinitesimal terms.
At any rate, then, the continuity of generality and particularity that he talks about in the Transcendental Dialectic, can be translated into a theory that the synthesis of a particular triangle, from the general scheme of triangulation, involves a continuous synthesis of the formal intuition of space in which we integrate Continuum-many infinitesimally divergent triangles in resolving to a given triangle in the formal intuition of space. Or, that is, logic relates its terms discretely, but imagination can relate them continuously, and carry down from logical generality (topic neutrality, so they say) schematic synthesis, in a continuous way. The similarity between the concept of triangles and the triangle scheme on the one side of things, and the triangle scheme and individual triangles on the other, is thus continuously gradable, and so it is not so much about "exact" similitude at any discrete moment, but an isomorphism of topological calculus, so to say, over a continuous range of moments.
†C.f. Kant's introduction of the categorical imperative, which has an "algebraic" flavor to it: we have the form of maxims as, "Adopt universal maxim x," and then we have to "solve for x" by stating the categorical imperative, yet maximal universality is the only part of the "equation" whose value we know, so it is the only grounds for determining what x is. So x ends up being a recursive/second-order representation of maximal universality, viz. this universality prescribes itself as an "end in itself."
Consider also when he says in the second Critique: When I subsume under a pure practical law an action possible to me in the world of sense, I am not concerned with the possibility of the action as an event in the world of sense. This is a matter that belongs to the decision of reason in its theoretic use according to the law of causality, which is a pure concept of the understanding, for which reason has a schema in the sensible intuition. Physical causality, or the condition under which it takes place, belongs to the physical concepts, the schema of which is sketched by transcendental imagination. Here, however, we have to do, not with the schema of a case that occurs according to laws, but with the schema of a law itself (if the word is allowable here), since the fact that the will (not the action relatively to its effect) is determined by the law alone without any other principle, connects the notion of causality with quite different conditions from those which constitute physical connection.
The physical law being a law to which the objects of sensible intuition, as such, are subject, must have a schema corresponding to it---that is, a general procedure of the imagination (by which it exhibits a priori to the senses the pure concept of the understanding which the law determines). But the law of freedom (that is, of a causality not subject to sensible conditions), and consequently the concept of the unconditionally good, cannot have any intuition, nor consequently any schema supplied to it for the purpose of its application in concreto. Consequently the moral law has no faculty but the understanding to aid its application to physical objects (not the imagination); and the understanding for the purposes of the judgement can provide for an idea of the reason, not a schema of the sensibility, but a law, though only as to its form as law; such a law, however, as can be exhibited in concreto in objects of the senses, and therefore a law of nature. We can therefore call this law the type of the moral law. [note that Kant was (probably) using the word "type" in the older sense of typology: not of types and tokens as nowadays, but of types and antitypes as referred to in some Christian theories of scriptural interpretation]