Editing Grothendieck topology (section)

=== Morphisms of sites ===
A continuous functor ''u'' : ''C'' → ''D'' is a '''morphism of sites''' ''D'' → ''C'' (''not'' ''C'' → ''D'') if ''u''<sup>''s''</sup> preserves finite limits.  In this case, ''u''<sup>''s''</sup> and ''u''<sub>''s''</sub> determine a geometric morphism of topoi <math>\tilde C \to \tilde D</math>. The reasoning behind the convention that a continuous functor ''C'' → ''D'' is said to determine a morphism of sites in the opposite direction is that this agrees with the intuition coming from the case of topological spaces.  A continuous map of topological spaces ''X'' → ''Y'' determines a continuous functor ''O''(''Y'') → ''O''(''X'').  Since the original map on topological spaces is said to send ''X'' to ''Y'', the morphism of sites is said to as well.

A particular case of this happens when a continuous functor admits a left adjoint. Suppose that ''u'' : ''C'' → ''D'' and ''v'' : ''D'' → ''C'' are functors with ''u'' right adjoint to ''v''. Then ''u'' is continuous if and only if ''v'' is cocontinuous, and when this happens, ''u''<sup>''s''</sup> is naturally isomorphic to ''v''<sup>*</sup> and ''u''<sub>''s''</sub> is naturally isomorphic to ''v''<sub>*</sub>.  In particular, ''u'' is a morphism of sites.