(Critique of Pure Reason §§9, 10, 13, 15)
In §9 of the Critique Kant argues that judgments differ from each other on four dimensions (quantity, quality, relation and modality) and on each dimension they can have three possible characteristics. For example on the dimension of 'relation', i.e. how it relates its parts to each other, a judgment can either be 'categorical ' (asserting a predicate of a subject), 'hypothetical' (asserting 'if .. then …') or 'disjunctive' (asserting 'either ... or ….'). This gives a total of 12 'functions of unity' in judgments.
In §10 he argues that in order to cognise an object I must not only be given a manifold of representations but must also synthesise (i.e. combine) these representations into a single whole using certain fundamental concepts. Furthermore I must use exactly the same faculty, namely the understanding, for combining representations so as to cognise an object as I do for combining concepts into a judgment. Therefore the same basic 'functions' of this faculty appear on the one hand as the functions of unity of judgments and on the other as the fundamental concepts through which objects are cognised, or what he calls the 'pure concepts of the understanding' or 'categories':
"The same function that gives unity to the different representations in a judgment also gives unity to the mere synthesis of different representations in an intuition, which, expressed generally, is called the pure concept of understanding. The same understanding, therefore, and indeed by means of the very same actions through which, in concepts, it produced the logical form of a judgment by means of analytic unity, also brings a transcendental content into its representations by means of the synthetic unity of the manifold in intuition in general, on account of which they are called pure concepts of the understanding that pertain to objects a priori ..." (g211-2, k112-3)
"Dieselbe Function, welche den verschiedenen Vorstellungen in einem Urtheile Einheit giebt, die giebt auch der bloßen Synthesis verschiedener Vorstellungen in einer Anschauung Einheit, welche, allgemein ausgedrückt, der reine Verstandesbegriff heißt. Derselbe Verstand also und zwar durch eben dieselben Handlungen, wodurch er in Begriffen vermittelst der analytischen Einheit die logische Form eines Urtheils zu Stande brachte, bringt auch vermittelst der synthetischen Einheit des Mannigfaltigen in der Anschauung überhaupt in seine Vorstellungen einen transscendentalen Inhalt, weswegen sie reine Verstandesbegriffe heißen, die a priori auf Objecte gehen ..." (A79-80, B104-105)
For example to the 'categorical' function of unity in judgments there corresponds the category of 'substance and accident', to the 'hypothetical' function there corresponds the category of 'cause and effect', and so on. So we can derive from the functions of unity in judgments a full list of 12 categories through which all objects must be cognised.What is the transcendental deduction for, then? In §13 Kant explains that we need a 'deduction' of the pure concepts of the understanding in order to allay the worry that, even though we can cannot but use the categories in order to combine our representations, nevertheless those representations as they are given to us might not be 'fit' for combining in this way. We can be sure that any manifold of sensible representations will be characterised by time and space because time and space are the pure forms of intuition, but we cannot be sure in the same way that any such manifold will be suitable for combining through the categories:
"[T]hat objects of sensible intuition must accord with the formal conditions of sensibility that lie in the mind a priori is clear from the fact that otherwise they would not be objects for us; but that they must also accord with the conditions that the understanding requires for the synthetic unity of thinking is a conclusion that is not so easily seen. For appearances could after all be so constituted that the understanding would not find them to be in accord with the conditions of its unity, and everything would lie in such confusion that, e.g., in the succession of appearances nothing would offer itself that would provide a rule of synthesis and so correspond to the concept of cause and effect, so that this concept would therefore be entirely empty, nugatory, and without significance. Appearances would nonetheless offer objects to our intuition, for intuition by no means requires the functions of thinking." (g222-3, k124)
"Denn daß Gegenstände der sinnlichen Anschauung den im Gemüth a priori liegenden formalen Bedingungen der Sinnlichkeit gemäß sein müssen, ist daraus klar, weil sie sonst nicht Gegenstände für uns sein würden; daß sie aber auch überdem den Bedingungen, deren der Verstand zur synthetischen Einheit des Denkens bedarf, gemäß sein müssen, davon ist die Schlußfolge nicht so leicht einzusehen. Denn es könnten wohl allenfalls Erscheinungen so beschaffen sein, daß der Verstand sie den Bedingungen seiner Einheit gar nicht gemäß fände, und alles so in Verwirrung läge, daß z.B. in der Reihenfolge der Erscheinungen sich nichts darböte, was eine Regel der Synthesis an die Hand gäbe und also dem Begriffe der Ursache und Wirkung entspräche, so daß dieser Begriff also ganz leer, nichtig und ohne Bedeutung wäre. Erscheinungen würden nichts destoweniger unserer Anschauung Gegenstände darbieten, denn die Anschauung bedarf der Functionen des Denkens auf keine Weise." (A90-91, B122-123)
Of course we find that our appearances are so constituted that the category of cause and effect applies to them, but Kant's point is that this is just an empirical fact. What we need is to show that any possible appearance must be so constituted that this and the other categories apply to it. This must be what he means by saying that we need to see how 'the subjective conditions of thinking' have 'objective validity' (g222, A89, B122). Showing that appearances are so constituted that the categories apply to them is the aim of the transcendental deduction. It is called a 'deduction' because in law a deduction means a proof of entitlement and if we can show that appearances must be so constituted that the categories apply to them then this will prove that we are rationally entitled to use them.
But this immediately raises a serious problem. What can Kant mean by the idea of an appearance 'corresponding to' a category? (I have tried to paraphrase this as the idea of the intuitions of the appearance being 'fit' for combining through the category, or the the idea of the category 'applying to' the appearance, but these reformulations don't help much.) He cannot mean that the appearance gives rise to (or is composed of) representations that are already somehow combined though that category, because Kant is emphatic that representations as they present themselves to us are not combined in any way. Combining them is something that we have to do, as he says in §15:
"[T]he combination (conjunctio) of a manifold in general can never come to us through the senses, and therefore cannot already be contained in the pure form of sensible intuition; for it is an act of the spontaneity of the power of representation […] we can represent nothing as combined in the object without having combined it ourselves …" (g245, k151)
"Allein die Verbindung (conjunctio) eines Mannigfaltigen überhaupt kann niemals durch Sinne in uns kommen und kann also auch nicht in der reinen Form der sinnlichen Anschauung zugleich mit enthalten sein; denn sie ist ein Actus der Spontaneität der Vorstellungskraft […] wir uns nichts als im Object verbunden vorstellen können, ohne es vorher selbst verbunden zu haben … " (B130)
So what else is it for an appearance to 'correspond to' a certain category? We need to know this before we can understand the aim of the transcendental deduction, which is just to prove that appearances must correspond to the categories.
(Edited: 23 July 2008)