An intuitively attractive way of packaging it is that Grothendieck topologies on a small category $C$ are in bijective correspondence with those operators $j: \Omega \to \Omega$ (where $\Omega$ is the truth value object in the topos $Set^{C^{op}}$) that satisfy the conditions $1_\Omega \leq j$, $j j = j$, and $j \circ \wedge = \wedge \circ (j \times j)$, (where $\wedge: \Omega \times \Omega \to \Omega$ is internal intersection). This is discussed in Mac Lane and Moerdijk for instance. Such meet-preserving closure operators form an inf-lattice, in fact a sub-inf-lattice of $\hom(\Omega, \Omega)$. The fact that $\hom(\Omega, \Omega)$ is an inf-lattice is easy because it is assembled as a limit of inf-lattices $\hom(h_c, \Omega) \cong \Omega(c)$ (the poset of sieves = subfunctors of the representable $h_c$), where infs are ordinary set-theoretic intersections. If one writes out the details of this, one is led to the proof in Borceux's book.
Added later: I suppose it is well-known that inf-lattices are automatically sup-lattices: the sup of a family is the inf of the set of upper bounds.