TITLE: What is the opposite category of $Set$?
QUESTION [21 upvotes]: In $Set$ the initial object is the empty set, and it has an unique morphism to each other object, namely $f=\emptyset$. However I find it difficult to think about the category ${Set}^{op}$, is there any morphism from $X \subset Obj(Set)$ to $\emptyset$ at all? Can an empty set be the codomain of some function?

REPLY [11 votes]: This is a pretty old question, so I'm a bit suprised that nobody mentioned the most simple answer. 
Since functions are the  right-unique, left-total binary relations, they reverses are the left-unique, right-total ones. So, $\mathsf{Set}^\mathrm{op}$ is the category with sets as objects and left-unique, right-total binary relations as morhisms.
Regarding relations as subsets of the Cartesian product, the $X\to \emptyset$ morphism is the empty set. This is really a morphism in $\mathsf{Set}^\mathrm{op}$ because it is a left-unique and right total binary relation.