Geometric morphism induced by a functor
= Geometric morphism induced by a functor
{title2=$f^*=(-)\circ F^{\mathrm{op}}$}
A functor $F:\mathcal C\to\mathcal D$ induces a <geometric morphism> from the presheaf topos on $\mathcal C$ to that on $\mathcal D$. Its inverse image is precomposition with $F^{\mathrm{op}}$, its direct image is <Right Kan extension> and its extra left adjoint is <left Kan extension>. The extra left adjoint sends $yC$ to $y(FC)$.