Suppose we are given a tripos $P$ over a category $C$; then we can form the category (in this case, a topos) $C[P]$ whose objects are pairs $(X,\approx)$ with $\approx$ a symmetric and transitive relation in the sense of $P$. Usually a terminal object in $C[P]$ is given by $(1,\top_{1\times 1})$, but in Realizability: An introduction to its categorical side the author notes that $(I,\top_{I\times I})$ works as a terminal object in $C[P]$ for any weakly terminal $I$.
My question is, what goes wrong, with respect to being a terminal object, if $I$ is not weakly terminal in $C$? My first thought was to try and find an example where this isn't terminal by taking $P$ to be the subobject tripos over $\mathbf{Set}^G$ for a group $G$, and $I$ to be the representable $G$-set, but since I'm not sure where the breakdown happens I'm not sure which conditions I should be poking at.