Given maps continuous maps $f:U \rightarrow X$ and $g:V \rightarrow X$ between topological spaces; then there is a unique map $f \sqcup g:U \sqcup V \rightarrow X$ whose restrictions to $U$ & $V$ are the original maps ie $(f \sqcup g)|U=f$ and $(f \sqcup g)|V=g$.
This is a categorical construction; take the pullback $f':U \times_X V \rightarrow U$, $g':U \times_X V \rightarrow V$; and then the pushout over that, and writing $U \sqcup V$ for this, we have by the unique mapping property of the pushout a unique map $f \sqcup g:U \sqcup V \rightarrow X$.
This works in any category with pullbacks and pushouts.
This generalises to more than two maps, I think, by using the wide pullback & pushout - Edit: I don't think it does.
Is there a standard name for this construction - taking the pushout over the pullback?