 Solution

Assume the diagonal $\Delta \subset X\times X$ is closed. For continuous $f,g:Y\to X$, the map $h:Y\to X\times X$, $h(y)=(f(y),g(y))$, is continuous: the inverse image of a basic rectangle $U\times V$ is $f^{-1}(U)\cap g^{-1}(V)$. Thereforeis closed as the inverse image of a closed set under a continuous map. This proves (ii) implies (iii).