Let $X$ be a set. To turn $X$ into a category $\mathcal{C}$ one is supposed to do the following: (1) take $\text{Ob}(\mathcal{C})=X$ and (2) for $x,y\in\text{Ob}(\mathcal{C})$ let $\mathcal{C}(x,y)=\{\text{pt}\}$ if $x=y$ and $\mathcal{C}(x,y)=\emptyset$ if $x\ne y$.
For each $x,y,z\in\text{Ob}(\mathcal{C})$, what is the composition law $$\mathcal{C}(x,y)\times\mathcal{C}(y,z)\rightarrow\mathcal{C}(x,z)$$ supposed to be? If there is at least one $\ne$ between the elements $x,y,z$, this mapping would have to be the empty mapping. However, in the case that $x=y=z$, is one supposed to write $$\text{pt}\circ\text{pt}=\text{pt}?$$ And in that case, would the "verification" of the associativity and identity axioms come down to the equations $$\text{pt}\circ\text{pt}=\text{pt}\ \land\ (\text{pt}\circ\text{pt})\circ\text{pt}=\text{pt}\circ(\text{pt}\circ\text{pt})?$$
What is this sham?–Surely I've made a mistake?
Yes, one is supposed to define $\mathrm{pt}\circ \mathrm{pt}=\mathrm{pt}$, and the axioms are indeed trivial to verify. This is not supposed to be an interesting example of a category, though it is a useful one.