Direct image with compact support

In mathematics, in the theory of sheaves the direct image with compact (or proper) support is an image functor for sheaves. It is one of Grothendieck's six operations. 
Definition.
Let "f": "X" → "Y" be a continuous mapping of topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on a topological space. The direct image with compact (or proper) support
sends a sheaf "F" on "X" to "f"!("F") defined by
where "U" is an open subset of "Y". The functoriality of this construction follows from the very basic properties of the support and the definition of sheaves.
Properties.
If "f" is proper, then "f"! equals "f"∗. In general, "f"!("F") is only a subsheaf of "f"∗("F")