Lemma 4.24.6. Let $u$ be a left adjoint to $v$ as in Definition 4.24.1.

Suppose that $M : \mathcal{I} \to \mathcal{C}$ is a diagram, and suppose that $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M$ exists in $\mathcal{C}$. Then $u(\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M) = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} u \circ M$. In other words, $u$ commutes with (representable) colimits.

Suppose that $M : \mathcal{I} \to \mathcal{D}$ is a diagram, and suppose that $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ exists in $\mathcal{D}$. Then $v(\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M) = \mathop{\mathrm{lim}}\nolimits _\mathcal {I} v \circ M$. In other words $v$ commutes with representable limits.

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