soft question - What's the notation for a function restricted to a subset of the codomain?

Concerning the name for the notion in question, but not the notation, Exposé 2 by A. Andreotti in the Séminaire A. Grothendieck 1957, available at www.numdam.org, suggests the following:

Consider a morphism $f:Arightarrow B$ in some category, subobjects $i:Urightarrow A$ and $j:Vrightarrow B$ of $A$ and $B$, respectively, and quotient objects $p:Arightarrow P$ and $q:Brightarrow Q$ of $A$ and $B$, respectively.

Then, $fcirc i$ is the restriction of $f$ to $U$. Dually, $qcirc f$ is the corestriction of $f$ to $Q$. (In particular, with the usual usage of the prefix "co", corestriction is not suitable for the notion in question.)

Moreover, if there is a morphism $g:Prightarrow B$ with $gcirc p=f$, then $g$ is the astriction of $f$ to $P$. Dually, if there is a morphism $h:Arightarrow V$ with $jcirc h=f$, then $v$ is the coastriction of $f$ to $V$. (Of course one can argue whether one should swap the terms astriction and coastriction (as suggested by Gerald Edgar).)

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