Left exact continuous functor defines morphism of sites

Corollary 6.21.

Let $(C,\tau)$ and $(D,\tau')$ be Grothendieck sites, and assume that the following two conditions are satisfied:

The category $$C$$ admits finite limits, and $u\colon C \to D$ preserves finite limits;
The functor $$u$$ preserves covering families.

Then $$u$$ is a continuous morphism of sites.

Proof

The left Kan extension functor

u_!\colon \PSh(C) \to \PSh(D)

is left exact by Proposition 2.43, as it is the colimit extension of the left exact functor

C \xrightarrow{u} D \hookrightarrow \PSh(D)

. Since

\(u\)

is a continuous functor by Lemma 6.20, the claim follows from Remark 6.19.