TITLE: Why is every category not isomorphic to its opposite?
QUESTION [32 upvotes]: This is a beginner category theory question:
I'm trying to wrap my head around the fact that we do not have $\mathbf{Sets} \cong \mathbf{Sets}^\mathsf{op}$, i.e., the category of sets is not isomorphic to its opposite.
As far as I have understood, an isomorphism between categories is a functor $F : \mathcal{C} \rightarrow \mathcal{D}$ for which there is an inverse $G : \mathcal{D} \rightarrow \mathcal{C}$ such that $FG = 1_\mathcal{D}$ and $GF = 1_\mathcal{C}$, i.e., an isomorphism in the category of categories.
Now, consider the functor $-^\mathsf{op}$, i.e., the functor sending its category to its opposite. Why is this functor not an isomorphism (for any category!), with itself as the inverse? I know that I have missed something, since this would imply that $\mathbf{Sets}$ is isomorphic to its opposite.

REPLY [2 votes]: The arguments above can be rephrased saying that ${\bf Sets}$ has a single initial objet (the empty set), and infinitely many final objects (all singleton sets), while for ${\bf Sets^{op}}$ this is all reversed.
An arrow of ${\bf Set}$ can be viewed as a family of sets indexed by the codomain, while in $\bf Set^{op}$, an arrow is a family of sets indexed by the domain. Still, the two categories are closely related, and would it not be legitimate to say that they are 'anti-isomorphic'?
In a debate between Bénabou, Cartier and others available on the web, it is pointed out that this makes the category ${\bf Set^{op}}$ "non-set-like", and more "topos-like". I didn't really understand, but this might be related to differences in the behavior of subobjects (which are quotient objects) and "truth values" in ${\bf Set^{op}}$.