Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram

Question

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 $f$?

Answer 1

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.