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Does anyone know whether there exist any "categorical version of deduction theorem"?
Lambek proved such a thing as part of his exploration of the connection between cartesian closed categories and intuitionistic logic. That result in particular can be found (which much related, as well as generalizations) in the early section of Lambek and Scott's "Introduction to Higher-Order Categorical Logic"
gershom, sayantan roy, a (<10) group began reading lambek & scott on monday last. (see FOM for the "would anyone like to read ..." note). we picked the 5pm (pdt) monday time, and initially zoom as the platform.