TITLE: What's the notation for a function restricted to a subset of the codomain?
QUESTION [27 upvotes]: Suppose I have a function f : A → B between two sets A and B. (The same question applies to group homomorphisms, continuous maps between topological spaces, etc. But for simpicity let's restrict ourselves to the case of arbitrary maps between sets.)
If I have subset A' ⊆ A of the domain A, then I can restrict f to A'. There is the function f|A' : A' → B which is given by f|A'(a) = f(a) for all a ∈ A'. 
Suppose now that I have subset B' ⊆ B of the codomain B that contains the image of the map f. Similarly I can restrict f to B', meaning that there is a function g : A → B' which is given by g(a) = f(a) for all a ∈ A. 
In general, it might be useful to consider such functions g and have a name for them, for example if I have a function B' → C that I want to apply afterwards. Unfortunately, I haven't seen a name or symbol for this function g in literature. 

Is there notation for the
  restriction g of f to a subset of the
  codomain similar to the notation
  f|A' for a restriction to a
  subset of the domain?

REPLY [3 votes]: Here's some notation.
Given a morphism $f : A \rightarrow B$ and a subobject $b': B' \rightarrow B$, the astriction of $f$ to $B' \subseteq B$ can be denoted $b'\backslash\, f : A \rightarrow B'$, pronounced "$b'$ under $f$." It satisfies $b'\circ (b'\backslash\, f).$ For example, if we have a function $f : A \rightarrow B$ and subsets $X$ of $A$ and $Y$ of $B$ such that $f(X)\subseteq Y$, then the relevant map $X \rightarrow Y$ can be denoted $Y\,\backslash\, (f \circ X).$
Similar notation can be used in linear algebra to denote solution sets. For example; to denote the set of all vectors $x$ satisfying a linear equation $Ax=b,$ I tend to write $A \backslash b.$ This has an intuitive look about it: $$Ax = b \iff x \in A \backslash b.$$
The connection basically is that in the category of matrices, $Ax=b$ can be seen as a factorization of $b$ through the tail of $A$ (because $A$ is on the left). So, the general principle is that:

In any category, $g \backslash \,f$ is good notation for the set of all factorizations of $f$ through the tail of $g$, or for the unique such factorization just in case a unique factorization exists.

Dually, if we have a function $f : A \rightarrow B$ and a quotient object $a' : A \rightarrow A'$, we can write $f/a'$ for the relevant morphism $A' \rightarrow B$ when it exists, or else use this notation to denote the set of all such morphisms.