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Tag 0039

Chapter 4: Categories > Section 4.24: Adjoint functors

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}
    \label{lemma-exact-adjoint}
    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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