The conceptual content of these results, when proved syntactically, is drowned in a swamp of symbolic detail which cannot be transferred to new situations. The remarkable construction which we use illustrates how much can be discovered simply by playing with adjoints and pullbacks. Since it preserves everything in sight, p 1 is also called a logical functor . A S be any functor. If U preserves this structure then so does p 0 .
L is full and faithful.
In sheaf theory this functor is traditionally called G , which of course conflicts with the notation of this book. 7.7.5 Q is full and faithful. U and Q preserve them. Corollary 7.7.13 shows that Q and A also preserve exponentials. We already know that the full subcategory of tight objects in S U is closed under definable limits, so the same is true of the class of tightly realised contexts.