Your understanding of transcendental deduction (TD) is much broader than how Kant actually employs the term in his Critique of Pure Reason (CPR). TD is a much smaller step in Kant's outline of his transcendental philosophy.
Briefly, in The Transcendental Deduction Kant wants to show that the "pure concepts of the understanding" (aka categories), which he derived from the logical forms of judgement in the previous chapter (The Metaphysical Deduction), actually apply to our experience.
The argument in TD runs as follows:
- A unified sensory experience can only possibly be accounted for if the categories apply to that sensory experience.
- [Insert complex/puzzling justification of claim (1) here…]
- Since we do perceive that kind of unified experience, the categories do in fact apply to our sensory experience.
With Hume against Hume
In order to understand Kant's philosophical move, TD must be set against Hume's empiricist psychological theory. Hume denied that apriori concepts could correctly apply to objects featured in experience, because only concepts (Hume: ideas) deriving from sensory experiences (Hume: impressions) can actually apply to the objects in our experience. Apriori concepts lack such a connection and therefore, according to Hume, fail to apply to the objects of experience - they lack objective validity.
Against this position, Kant employes TD to prove that certain apriori concepts do correctly apply to objects featured in our experience, i.e. that such apriori concepts have objective validity after all. In doing so, Kant doesn't counter Hume's point, he actually concedes that apriori concepts cannot be derived from sensory experiences. He does, however, object to Hume's conclusion that a priori concepts therefore cannot apply to the objects of experience at all. To block this conclusion, Kant needs to show that there's another, non-empirical way in which apriori concepts can correctly apply to experience and achieve objective validity. This alternative way is through a transcendental argument, see the outline above.
This is why Kant employes a deduction. Kant's "deduction" is not used in the modern sense, but in a way used in German legal language at the time. Here Deduktion meant a proof of the quid iuris (the matter of the law) in opposition to the quid facti (the matter of the fact). As Kant's argument does not involve the rule of law in a literal sense, the analogy that he tries to convey is about a legitimate source of justification not concerned with establishing matters of fact.
Because you write that
the role of cognition and sense is to verify whether these categories apply to objects
it is important to stress that this is exactly not Kant's point. The opposite is the case: TD itself is the verification! – The "proof" that his categories apply to the objects of experience is exactly what Kant tries to achieve through TD.
Kant's argument is not about cognition in any "naturalized" sense of the term. Kant is not interested in asserting an empirical hypothesis that might be somehow verified by empirical means (that would have been an "empirical deduction", as Kant calls it, which Hume already showed to fail for apriori concepts).
To get an overview of the actual argument deployed in TD and also get a better grasp on how TD fits into Kant's broader project, you may read a very good summary from a philosophy course at UC Davis.