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 of 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 of 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 a 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?