Kan extension
= Kan extension
{title2=$\operatorname{Lan}_f,\quad\operatorname{Ran}_f$}
A <left Kan extension> is <left adjoint> to precomposition, when it exists; a <Right Kan extension> is <right adjoint>. For small indexing categories and set-valued <functors>, both exist by comma-category <colimit> and limit formulas. These two adjoints give the <essential geometric morphism> induced by a <functor> between presheaf indexing categories.