The following question is taken from $\textit{Arrows, Structures and Functors the categorical imperative}$ by Arbib and Manes
Exercise: Let $X$ be a vector space and let $\leq$ be the partial ordering on $X$ of exercise 1. Verify that the following data define a category $\textbf{K}.$ The objects of $\textbf{K}$ are the elements of $X$. For $x,y$ in $X,$ $\textbf{K}(x,y)=\{\lambda\mid \lambda x\leq y\}.$ Composition is ordinary multiplication. Observe that there are many pairs of objects $(x,y)$ such that $\textbf{K}(x,y)=\emptyset.$
I would like to know how composition works, since to show that the object $\textbf{K}$ is a category, i have to use composition of morphism.
Is the composition of morphism for the mapping $\textbf{K}(x,y)\times \textbf{K}(y,z)\rightarrow \textbf{K}(y,z)$ defined as $\lambda x\leq z$ or ${\lambda}^2 x\leq z$?, where we have $\textbf{K}(x,y)=\{\lambda\mid \lambda x\leq y\}$ and $\textbf{K}(y,z)=\{\lambda\mid \lambda y\leq z\}.$ Also, how does one get $\textbf{K}(x,y)=\emptyset?$
Thank you in advance