I am now working on Munkres Topology,Stone–Čech compactification part. He says that the correspondence between a completely regular space and its Stone–Čech compactification is a funtor. To verify this, I need to show that the correspondence preserves the identity mapping and composites of functions. It was easy to show the former, but I am not sure how to do the latter... The situation is:
Let $\beta(X)$ denote a Stone–Čech compactification of a topological space $X$.
Let $X,Y,Z$ be completely regular spaces.
Let $f:X\rightarrow Y$ , $g:Y\rightarrow Z$ be continuous maps.
Let $\beta(f):\beta(X) \rightarrow \beta(Y)$ extend $\iota \circ f$, where $\iota:Y \rightarrow \beta(Y)$ is an inclusion mapping.
What I need to show is that
$$\beta(g\circ f)=\beta(g) \circ\beta(f)$$
How can I show this? It seems obvious if $x\in X$. But how can I show for the case $x\in \beta(X)-X$? Any help would be really appreciated. Thanks.