**Definition:** Let $X$ and $Y$ be two topological spaces. A continuous map $f: X \rightarrow Y$ is called **proper** if $f^{-1}(K)$ is compact for every $K \subset Y$ compact.

**Question:** If I take the category where objects are locally compact Hausdorff spaces and morphisms are proper maps, does this really form a category?

I think the identity morphism for any object would be just identity proper map, which seems clear. However, I am not sure about the associativity of the composition. For three objects If $f: A \rightarrow B$, $g: B \rightarrow C$ and $h: C \rightarrow D$, where $f, g, h$ are proper maps and $A, B, C, D$ are objects in the category, will $(h \circ g) \circ f = h \circ (g \circ f)$?