If I have a function $f: X \to Y$ and a subset $A \subseteq X$ then I can define the restriction $f|_{A}: A \to X$ by $f|_{A}(a) = f(a)$ for all $a \in A$. This can be interpreted as composing with the inclusion map $\iota: A \to X$ given by $\iota(a) = a$ for all $a \in A$ since we have $f|_{A} = f \circ \iota$, and interpreting $f|_{A}$ in this way can be quite useful (for example because the inclusion map is continuous in topology and a homomorphism in algebra, and so if $f$ is continuous/a homomorphism, so is $f|_{A}$ since it is a composition of continuous functions/homomorphisms).
However what if instead I have a superset $B$ containing $Y$? Let's say I call my altered function $f_B^{\ast}: X \to B$ so that $f_B^{\ast}(x) = f(x)$ for all $x \in X$. I can similarly define the inclusion map $\iota: Y \to B$ and then I have $f_B^{\ast} = \iota \circ f$. All I am doing is increasing the codomain to include values which don't get mapped to. For example any function into the natural numbers can equally be considered a function into the integers, or the rationals, or reals, complex numbers, quaternions, and so on. I feel like this is something mathematicians do without reflecting it in their notation. However to me it seems very much dual to the restriction of a function, and I was wondering if it has a name. Thanks for taking the time to read.
To answer my own question, see here: https://mathoverflow.net/questions/29911/whats-the-notation-for-a-function-restricted-to-a-subset-of-the-codomain (I have no idea why I couldn't find this when I was Googleing for answers). What I'm talking about is indeed dual to the usual restriction, and so the obvious name for it is the "corestriction".