I wish to prove the following claim: Let $\tau$ be the initial topology on $X$ induced by the family of maps $\{f_i: X\rightarrow Y_i\}, i\in I$. Then a map $g: Z\rightarrow X$ is continuous if and only if $f_i\circ g: Z\rightarrow Y_i$ is continuous for all $i\in I$.
I'm having trouble with the opposite direction. I would really appreciate some help, thank you!
