I'm having a question related to this thread: Show that the path components $\pi_0$ of $X = U \cup V$ form a pushout diagram.
Both the original post and the suggested answers seem to agree that the maps $f$, $g$, $F$, $G$ that send an element to its path component would make the diagram commute. And indeed they do. But in the definition of pushout that I read, the maps $f$, $g$, $F$ and $G$ are usually given, and taken to be some general maps. Wouldn't explicitly specifying them sort of affects the generality of the diagram? Or is it the case here that we're arguing that the maps $f$, $g$, $F$ and $G$ defined that way would turn the diagram into a pushout, but they do not exclude the possibility that some other maps might also work?
Saying a diagram is a pushout diagram is asserting that it is a cone over a pushout diagram along with the pushout diagram. In this case the portion $ A \xleftarrow{g} B \xrightarrow{f} C$ is the pushout diagram while the $A \xrightarrow{F} D \xleftarrow{G} C$ is the cone.