$X,Y,Z$ are topological spaces and $X,Y$ are Hausdorf.
Prove the homeomorphism over the compact open topology $Map(X \amalg Y,Z) \cong Map(X,Y)\times Map(Y,Z)$
I know that $(f: X \amalg Y \rightarrow Z) \mapsto (f \circ i_{1}, f \circ i_{2})$ where $i_{1},i_{2}$ are the canoncial inclusions.
The universal property tells us that this is well-defined and a bijection. I'm unsure how to further prove this homeomorphism.