How can I understand maps out of a limit or into a colimit? If $L$ is a limit, we know that maps $A\to L$ are characterized by cones from $A$. Similarly if $C$ is a colimit, maps $C\to B$ are characterized by cocones into $B$.
Is there a general way to understand maps out of a limit $L\to A'$ or maps into a colimit $B'\to C$?
Or is the only way to work with such maps given by the specific category I am working with?

The motivating example for this is that I have a colimit $C$ of the diagram
$$B_0\to B_1\to B_2\to \ldots$$
of simplicial sets, and given that for each $i$ there is a pullback square
$$
\require{AMScd}
\begin{CD}
B_0@>>> B_i   \\
@VVV  @VVV   \\
A @>>> X
\end{CD}
$$
I want to show that if I replace $B_i$ with $C$ this is also a pullback square.
I am thinking that given a simplex $f:\Delta_n\to C$, I should be able to factor it through some $B_i$, but I'm having trouble showing this.
 A: You can't understand maps into colimits or out of limits in general, no. But there are various interesting special cases in which you can. For instance, a compact object in a category $C$ is an object $X$ such that maps out of $X$ commute with filtered colimits, that is, $C(X,\mathrm{colim}Y_i)\cong \mathrm{colim}C(X, Y_i)$ whenever $Y_i$ is a filtered system in $C$, such as a sequence $Y_i\to Y_{i+1}$, as you have in your question. The representables in any presheaf category are compact. Indeed, they're "small-projective" in the sense that maps out of them commute with arbitrary colimits, a very rare property indeed. 
The other notable reason you might be able to characterize maps into a colimit is if it's also a limit! This is very common for absolute colimits: splittings of idempotents, biproducts in additive categories, and suspensions in differential graded categories are good examples. 
One explanation of why this is so rare is that a colimit is already characterized up to isomorphism. If you know the maps both into and out of an object, then you've characterized it up to isomorphism in two distinct ways, which have to agree-and this is hard to do.
