Is the function $\mbox{Rings}\rightarrow\mbox{Sets}$ given by $R\mapsto \{\pm 1\in R\}$ corepresentable?
Of course this might be problematic in characteristic 2 since this set is then a singleton, but a char-2 ring can't map to anything but a char-2 ring anyways, so maybe it's alright. Actually what I was originally thinking about is the cogroupoid-in-$\mbox{Rings}$ $(A,\Gamma)$ corepresenting the groupoid whose objects are $\{x^2+bx+c:b,c\in R\}$, with morphisms Hom((b,c),(b',c')) = \{ r \in R : (x+r)^2+b(x+r)+c = x^2+b'x+c\}. Explicitly, $A=\mathbb{Z}[b,c]$ and $\Gamma = A[r]$ (where the copy of $A$ selects the source of a morphism). I'd like to extend this to allow my morphisms to take the form $x\mapsto \pm x + r$ (instead of just $x\mapsto x + r$, as it is now). The obvious guess is to set $\Gamma = A[e,r]/((e-1)(e+1)) = A[e,r]/(e^2-1)$, but of course this is going to allow our linear coefficient to be any order-2 element of $R^\times$.
I suspect there isn't such a ring, because I'd think if there were then it'd be easy to see what it should be. Either way, there's probably an algebro-geometric reason for the answer, which I'd love to see.
EDIT: To be completely clear: all my rings are commutative and have 1, and all my ring homomorphisms are unital.