# The Stacks Project

## Tag 0039

Lemma 4.24.5. Let $u$ be a left adjoint of $v$ as in Definition 4.24.1.

1. If $\mathcal{C}$ has finite colimits, then $u$ is right exact.
2. If $\mathcal{D}$ has finite limits, then $v$ is left exact.

Proof. Obvious from the definitions and Lemma 4.24.4. $\square$

The code snippet corresponding to this tag is a part of the file categories.tex and is located in lines 3198–3205 (see updates for more information).

\begin{lemma}
Let $u$ be a left adjoint of $v$ as in Definition \ref{definition-adjoint}.
\begin{enumerate}
\item If $\mathcal{C}$ has finite colimits, then $u$ is right exact.
\item If $\mathcal{D}$ has finite limits, then $v$ is left exact.
\end{enumerate}
\end{lemma}

\begin{proof}
Obvious from the definitions and Lemma \ref{lemma-adjoint-exact}.
\end{proof}

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