For any relation $R$, both the domain and range are logically computational. A codomain of this relation is any superset of its range. Similarly, a predomain is any superset of its domain. We can then say $R:X\to Y$ where $X$ and $Y$ are any predomain and codomain.
A category is a class of "objects" equipped with a class of "morphisms" or "arrows" between those objects. For concrete categories (specifically the ones whose morphisms are functional), morphisms are functions $f:X\to Y$ where $X$ and $Y$ are the domain and codomain of $f$.
My question is: is there some other structure to $f$ which encodes a specific codomain? Otherwise it seems that this function would either exist as all arrows from $X$ to each of its codomains in the category or as precisely one arrow from $X$ to its unique range.
Maybe $(f,X,Y)$ or other variation is the structure of a morphism in a concrete category?
If I understand what you're asking, this comes down to a convention issue: do we think of a relation as just a set of ordered pairs or as a set of ordered pairs together with a choice of predomain and codomain? Similarly, do we think of a function as a set of ordered pairs or as a set of ordered pairs together with a choice of codomain?
Some mathematicians, e.g. set theorists, tend to take the former tack. It seems you're working along these lines as well. In that case we will indeed often want our morphisms to be not functions but "functions + codomain data."
However, other mathematicians consider the predomain/codomain data to be part of the relation in the first place. Indeed, in my experience this is more common. In this case there's no "missing data" issue.
So your concern is absolutely valid, but you may not see it highlighted in practice since it may be obviated by the use of different definitions.