Inverse image functor of a geometric morphism
= Inverse image functor of a geometric morphism
{title2=$f^*:\mathcal F\to\mathcal E$}
= Inverse image functors of geometric morphisms
{synonym}
The inverse image is a finite-limit-preserving <left adjoint> with <right adjoint> $f_*$. 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.