Consider the direct limit of an indexed family $\{a\_n\}\_{n\in \omega}$:


$\require{AMScd}$ 
\begin{CD}
a\_0 @>>> \ldots @>>> a\_n @>>> a\_{n+1} @>>> \ldots\\
@Vi\_0VV \ldots @. \ldots @Vi\_nVV @Vi\_{n+1}VV \ldots \\
a @= \ldots @= a @= a @= \ldots
\end{CD}


Let $f:a \to b$ be an arrow from $a$ to $b$.
We can map $f$ to the family of arrows $\{f\_n\}\_{n\in\omega}$
where $f\_n = f\circ i\_n$.


E.g. think of $\mathbb{R}^\omega$ as the direct limit of $\{\mathbb{R}^n\}\_{n\in\omega}$ with arrows 
$(x\_0, x\_1, \ldots, x\_n) \mapsto (x\_0, x\_1, \ldots, x\_n, 0)$
and a function $f$ from $\mathbb{R}^\omega$ to $\mathbb{R}$.


I am looking for a name to refer to this construction.
Is there a general name for the functor that maps $f$ to $\{f\_n\}\_{n\in\mathbb{N}}$?
(or some specific instance of this functor e.g. in differential geometry)?
Would it be reasonable to refer to the family $\{f\_n\}\_{n\in\mathbb{N}}$ as "sections" of $f$?



Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram
The composition of the cone over $a$ with $f$.


In case you are asking how to call the passage from $f : (\sum\_i A\_i) \to B$ to $(f\_i : A\_i \to B)\_i$, that would be "precomposition with the injections $A\_j \to \sum\_i A\_i$" but you will not like the answer because you are looking for an "established" term. Apparently, category theory is not established enough ;-)


If you do not want to educate your audience and teach them bits of category theory (or type theory) then you could simply call this process **decomposition of $f$**. I suspect "ordinary" mathematicians do not have an established phrase for a very simple reason: ordinary mathematicians do not take coproducts of sets seriously.


