Image functors for sheaves

In mathematics, especially in sheaf theory—a domain applied in areas such as topology, logic and algebraic geometry—there are four image functors for sheaves that belong together in various senses.
Given a continuous mapping "f": "X" → "Y" of topological spaces, and the category Sh(–) of sheaves of abelian groups on a topological space. The functors in question are
The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called ""f" shriek" or ""f" lower shriek" and ""f" upper shriek"—see also shriek map.
The exceptional inverse image is in general defined on the level of derived categories only. Similar considerations apply to étale sheaves on schemes.
Adjointness.
The functors are adjoint to each other as depicted at the right, where, as usual, formula_1 means that "F" is left adjoint to "G" (equivalently "G" right adjoint to "F"), i.e. 
for any two objects "A", "B" in the two categories being adjoint by "F" and "G".
For example, "f"∗ is the left adjoint of "f"*. By the standard reasoning with adjointness relations, there are natural unit and counit morphisms formula_2 and formula_3 for formula_4 on "Y" and formula_5 on "X", respectively. However, these are "almost never" isomorphisms—see the localization example below.
Verdier duality.
Verdier duality gives another link between them: morally speaking, it exchanges "∗" and "!", i.e. in the synopsis above it exchanges functors along the diagonals. For example the direct image is dual to the direct image with compact support. This phenomenon is studied and used in the theory of perverse sheaves.
Base Change.
Another useful property of the image functors is base change. Given continuous maps formula_6 and formula_7, which induce morphisms formula_8 and formula_9, there exists a canonical isomorphism formula_10.
Localization.
In the particular situation of a closed subspace "i": "Z" ⊂ "X" and the complementary open subset "j": "U" ⊂ "X", the situation simplifies insofar that for "j"∗="j"! and "i"!="i"∗ and for any sheaf "F" on "X", one gets exact sequences
Its Verdier dual reads
a distinguished triangle in the derived category of sheaves on "X".
The adjointness relations read in this case
and