Functor between $\textbf{Rel}$ and $\textbf{Rel}^{\text{op}}$.

Question

This is a reference to Awodey's Category Theory Exercise 1.9 (1c). I need to define a functor $C:\textbf{Rel}\longmapsto \textbf{Rel}^{\text{op}}$ such that $C(R:A\times B)=\{(b,a):(a,b)\in R\}=:R^{*}:B\times A$. The first thing that occurs to me is to define $C$ by $C(A):=A$ and $C(R:A\times B):=R^{*}:B\times A$, but I think it'll be a problem since $C$ needs to satisfy $C(R:A\times B)=C(R):C(A)\times C(B)$, which is different from $R^{*}:B\times A$. Is it ok to define it like this?