# The Stacks Project

## Tag: 03DC

This tag has label sites-modules-lemma-exactness-pushforward-pullback and it points to

The corresponding content:

Lemma 17.14.3. Let $f : (\mathop{\textit{Sh}}\nolimits(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\mathop{\textit{Sh}}\nolimits(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
1. The functor $f_*$ is left exact. In fact it commutes with all limits.
2. The functor $f^*$ is right exact. In fact it commutes with all colimits.

Proof. This is true because $(f^*, f_*)$ is an adjoint pair of functors, see Lemma 17.13.2. See Categories, Section 4.23. $\square$

\begin{lemma}
\label{lemma-exactness-pushforward-pullback}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to (\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$
be a morphism of ringed topoi.
\begin{enumerate}
\item The functor $f_*$ is left exact. In fact it commutes with
all limits.
\item The functor $f^*$ is right exact. In fact it commutes
with all colimits.
\end{enumerate}
\end{lemma}

\begin{proof}
This is true because $(f^*, f_*)$ is an adjoint pair
of functors, see
\end{proof}


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