When proving that the subobject classifier of a topos is an internal Heyting algebra, we exploit the natural isomorphism $$Sub(X)\simeq Hom(X,\Omega)$$ Therefore the intersection of subobjects induces an arrow $$Hom(X,\Omega)\times Hom(X,\Omega)\rightarrow Hom(X,\Omega)$$ and so by Yoneda lemma and the isomorphism $Hom(X,\Omega\times \Omega)\simeq Hom(X,\Omega)\times Hom(X,\Omega)$ we get $$\wedge:\Omega\times\Omega\rightarrow\Omega$$ The same strategy yields all other arrows which define a Heyting algebra object structure for $\Omega$. A similar strategy yields arrows for every power object $PX$. (Ref. MacLane Moerdijk around page 188)
What I'm curious about is the following: is there any known result about inducing algebra/lattice structures on particular subobjects of $PX$?
For example, suppose known a particular subset $u(X)\subseteq Sub(X)$ which happens to be closed under intersection: can we deduce that there is a subobject $U(X)\rightarrowtail P(X)$ with its own intersection arrow, compatible with all the necessary structure? Is there any way to build it if we cannot use Yoneda anymore?