Fix a set $I$ and consider the category $\mathbf C$ where the objects are the maps $x:X\to I$ in $\mathbf {Set}$, and the morphisms are the maps $f:X\to Y$ such that $x=yf$. What is a subobject classifier for $\mathbf C$? The only way thing that came to my mind was to define the special object as $\pi_1:I\times I\to I$ (the projection on the first copy of $I$), and to choose $\langle 1_I,1_I\rangle$ as the map from the terminal object. Given a monomorphism $h:S\rightarrowtail X$ (s.t. $s=hx$), if I chose as "characteristic function" $\langle x,x \rangle :X\to I\times I $, the diagram obtained is commutative but I don't know if it's a pullback; honestly it doesn't seem a subobject classifier in any way to me. Can you just give me a hint to find the special object, to start? Thank you
P.S. I don't know how much standard it is, but the notation $\langle a,b\rangle : C\to A\times B$, given arrows $a:C\to A$, $b:C\to B$, is the map defined using the universal property of the product.
Hint 1: There is an equivalence of categories $\mathbf{Set}/I \simeq \mathbf{Set}^I$.
Hint 2: The subobject classifier in a product of categories with subobject classifiers is easy to describe.