Cognitions a priori are the intellect's own forms, because a priori means not come from outside, what is in the intellect without coming from outside is its own, and what is its own and the manner its objects must present themselves is its forms of cognizing

The passage

Because a priori means not gained by way of experience and so not come into us from outside, what is present in the intellect without having come from outside belongs to it originally as its own essence, and what belongs to it originally and consists in the universal manner in which all its objects must present themselves to it is the forms of its cognizing, the part of our cognitions we are conscious of a priori, which the text takes as a fact proved by Kant, is the forms of the intellect's cognizing, given that these cognitions consist in that universal manner (the map's bridge, discharging the text's conditional), and so »cognitions a priori« and »the intellect's own forms« are two expressions for the same thing, since two expressions of which the one names what the other names are expressions for one thing (the map's bridge for the text's »Demnach«).

The argument

1. P1We are conscious of a part of our cognitions a priori, a fact strictly proved by Kant
2. P2A priori means nothing other than not gained by way of experience, not come into us from outside
3. P3What is present in the intellect without having come from outside is what belongs to it originally, its own essence
4. P4If what is thus present in the intellect consists in the universal manner in which all its objects must present themselves, then these are the forms of its cognizing
5. P5Assumption. The a priori cognitions, the part of our cognitions we are conscious of a priori, consist in the universal manner in which all the intellect's objects must present themselves to it (the map's bridge: the antecedent of the text's conditional, which its »Demnach« treats as discharged without asserting it; what the Aesthetic's theorems say of space and time, and no sentence of this paragraph asserts in so many words)
6. P6Assumption. If the a priori cognitions are the forms of the intellect's cognizing, then »cognitions a priori« and »the intellect's own forms« are two expressions for the same thing (the map's bridge for the text's »Demnach«)
7. Infer.hypothetical syllogism and modus ponens
8. C»Cognitions a priori« and »the intellect's own forms« are two expressions for the same thing

The argument is the paragraph at ZA II 536–537 from its fourth sentence, »Denn«, to its last, »Demnach«: a priori means not come from outside; what is in the intellect without having come from outside is its own; if what is thus in it is the manner all its objects must present themselves, it is the forms of its cognizing; accordingly cognitions a priori and the intellect's own forms are two expressions for one thing.

What the form captures and what it leaves out. Premise 1 is the fact the third sentence takes as proved by Kant (c-schopenhauer-part-of-our-cognitions-a-priori-a-fact-proved-by-kant), that a part of our cognitions is a priori; the atoms speak of that part throughout. Premise 2 is the definition (c-schopenhauer-a-priori-means-not-gained-through-experience), a → ¬e; the text's »also« joins »nicht auf dem Wege der Erfahrung gewonnen« and »nicht von außen in uns gekommen«, and the atom e carries both. Premise 3 is the fifth sentence (c-schopenhauer-what-is-in-the-intellect-without-coming-from-outside-is-its-own-essence), ¬e → b. Premise 4 is the sixth sentence's conditional (c-schopenhauer-if-what-is-in-the-intellect-is-how-objects-must-present-themselves-it-is-its-forms), (b ∧ m) → f, where b glosses »dies so in ihm selbst Vorhandene«, what the sentences before have reached. The first bridge, m, is the map's: the text's conditional is never discharged in so many words, and its »Demnach« draws the conclusion as if it had been; that the a priori cognitions consist in the universal manner all objects must present themselves to the intellect is what the Aesthetic's theorems say of space and time, which the paragraph's second sentence counts among the irrefutable truths (c-schopenhauer-theorems-of-transcendental-aesthetic-irrefutable-truths), and the map names the step as its own rather than read it into that sentence. The second bridge, f → s, is the map's for »Demnach«: if the a priori cognitions are the forms of the intellect's cognizing, the two expressions name one thing. Both are named in the statement. The form proves s, the conclusion claim's proposition, its two hedges, »im Grunde nur« and »gewissermaaßen«, left to the statement (c-schopenhauer-cognitions-a-priori-and-forms-of-the-intellect-two-expressions-for-one-thing); it does not prove the third sentence's »gar keine andere Erklärung« (c-schopenhauer-a-priori-cognitions-explained-only-as-forms-of-the-intellect), which the paragraph states as its thesis and which the chain supports without establishing the uniqueness. The check says nothing of whether any cognition is a priori, nor of whether Kant proved it; what the map holds of Kant's own proof for space is A 24 / B 38 (c-kant-space-a-priori), and the comparison x-thing-in-itself says why no link is set between that and this general conclusion.