I have a very common situation, for which I need both: (1) notation; and, if available, (2) a general relative term. Let's say that:
- there is a functor between categories, $f:C_1\to C_2$,
- $c_1$ is a particular object of $C_1$
- $c_2$ is a particular object of $C_2$, such that in mapping $C_1$ to $C_2$, $f$ maps $c_1$ to $c_2$
What is the name given to a morphism that maps $c_1$ to $c_2$ in the same way that $f$ does, independently of the existence of $C_1$ or $C_2$? What concise notation can I use to refer to such a morphism?
I am sure that there is a clear answer for this, but so that the point of my confusion is more clear, these are my intuitions about the issue:
$f$ itself can't be the answer, because it is a specialization of the morphism that I am referring to, because it communicates a lot more information than just $c_1\to c_2$. For example, it is possible that another functor could map two completely different categories and still meet the criteria of mapping $c_1$ to $c_2$. (e.g. if $g: C_3 \to C_4$ could map $c_1$ to $c_2$ despite being distinct from $C_1$ and $C_2$)
$f(c_1)$ seems closer to what I am looking for, but I think that $f(c_1)$ should actually refer to the resultant value, or $c_2$ itself. I am interested in the morphism between $c_1$ and $c_2$, rather than only $c_2$.
Update: If my question seems nonsensical for reasons raised by both @Jim and @AlexKruckman, let's just consider $c_1$ and $c_2$ to be categories themselves and the morphism I am asking about to also be a functor itself.