Inverse image functor of a geometric morphism ($f^{\ast }:\mathcal{F}\to \mathcal{E}$)

The inverse image is a finite-limit-preserving left adjoint with right adjoint $f_{\ast }$. It preserves arbitrary colimits, and hence images as well as finite limits. These properties preserve the interpretation of geometric formulas. It need not preserve Heyting implication or every universal quantifier.