How do we show that the category of sets is not equivalent to it's opposite. I know we use $\emptyset$ and singletons to show that they are not isomorphic.
The definition of equivalence that I am using is that there are two functors $F:sets\rightarrow sets^{op}$ and $G:sets^{op}\rightarrow sets$ such that $FG$ is naturally isomorphic to $1_{sets}$ and $GF$ is naturally isomorphic to $1_{sets^{op}}$
Well, there are lots of non-invertible morphisms $1 \to X$. But every morphism $X \to \emptyset$ is invertible. Hence $\mathbf{Set}$ is not equivalent to $\mathbf{Set}^\mathrm{op}$ either.