What can be said about the terminal object in a category of pullbacks? Given a category $A$; consider the category of arrows $A^2$, whose morphisms are commutative squares, which are further pullback squares. Suppose this category has a terminal object $a \rightarrow b$. What can be said about it? For example is $a$ terminal in $A$?
 A: Such an arrow is called an object classifier. See this nlab article (in the link, see the part beginning "When $S$ is the class of all morphisms..."), which treats a more general notion, but mentions this case. The article mentions that object classifiers typically don't exist in interesting categories for size reasons.
 I am pretty sure that if $A$ has finite limits and $a \to b$ is a subobject classifier in $A$, then $a$ must be terminal (a similar fact is alluded to in the last sentence of the "Definition" section of subobject classifier on the nlab). An object classifier is in particular a subobject classifier, so if $A$ has finite limits and an object classifier, you are correct that $a$ must be terminal. 
But this leads to some hints about why object classifiers are rare. If $a \to b$ is an object classifier and $a$ is terminal in $A$, then $a \to b$ is monic. Since monics are stable under pullback, every arrow in $A$ is monic! I can't think of any example of a category with a (sub)object classifier and all arrows monic which is not equivalent to the terminal category.
EDIT
On reflection, I now believe that "if $A$ has finite limits and a subobject classifier, then the domain of the subobject classifier is terminal" requires you to take being a monomorphism as part of the definition of a subobject classifier. So some of my logic is circular. But it's still true that if the domain of your object classifier is terminal, then every arrow is mono.
