Lemma 4.24.5. 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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