Terminal object in $Set^{C^{op}}$ and subobject classifier. This is from Sheaves in Geometry and Logic pg 38.
I'm not sure if I understood it correctly but the subobject classifier in  $Set^{C^{op}}$ when $C$ is a small category is a map (natural transformation) which sends every object $D$ to the maximal sieve on $D$
so we've got $ true : 1 \implies \Omega$ .
What is $1$ in this category? I thought if $C^{op}$ has a terminal object $1_{c^{op}}$ then  i could consider $Y(1_{^{op}})$ but i have no idea.
 A: The terminal object is in fact the constant functor to a fixed one-point set. This can be seen from the fact that the value $F(C)$ at any object $C\in\mathscr{C}$ has a unique map to the terminal object in $\mathbf{Sets}$, and obviously the terminal object restricts along all morphisms in $\mathscr{C}$. $\Omega (C)$ returns the collection of all sieves on $C$ (or in a sheaf category - the collection of all $J$-closed sieves - sieves which contain every object that they $J$-cover).
Truth then corresponds to the natural transformation taking each $\{*\}$ to the maximal element of $\Omega (C)$. This corresponds to the fact that the section in question lives on ``all of $C$''. In general, for a given section on $F\in \text{Sh}(\mathscr{C})$ at $C$, the truth value associated with that section corresponding with a subsheaf $G$ will be the largest closed sieve on $C$ such that the section lives on the whole region specified by the sieve in $G$. In the extreme case, when it is on all of $C$, this will be the maximal sieve corresponding to $C$ itself (if you want to think in terms of a fixed site).
