Consider a function $f : A \rightarrow B$.
We can restrict the function to any subset of its domain. That is completely standard.
Sometimes we would also like to restrict the codomain of a function. Notably and obviously, the codomain must be large enough to contain the image $f(A)$ of the domain under $f$.
Is there a word or a formalism for restricting the codomain of a function?
Given a function $f : A \to B$ and a subset $V \subseteq B$ such that $f[A] \subseteq V$, the function $A \to V$ agreeing with $f$ is called the corestriction of $f$ to $V$.
The term is commonly used by category theorists such as myself (see here for example) since corestriction is quite literally the categorical dual of restriction.
See also this related question on MathOverflow.