So I basically have the following. Let $OP(-):\textbf{Top}\rightarrow \textbf{Cat}$, be given by $X\mapsto OP(X)$ where $OP(X)$ is defined as the category whose objects are open subsets $U$ of $X$, and whose morphisms are defined as $Mor(U,V)=\{incl_{VU}\}$ if $U\subset V$ and $\emptyset$ otherwise.
The problem is asking to show that $OP(-)$ is a contravariant functor. I have proven that indeed $OP(X)$ is a small category, so all I need to prove is that if $f$ is a continuous function from a topological space $X$ to $Y$, then $OP(f)$ should be a morphism (which in $\textbf{Cat}$ is a functor) from $OP(Y)$ to $OP(X)$. The problem is that the problem doesnt specify what $OP(-)$ does to the morphisms of $\textbf{Top}$. I dont know if there is a natural way to define what $OP(-)$ does to the arrows of $\textbf{Top}$. Any help would be greatly appreciated.
Thanks.