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The Stacks project

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

  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.

  2. 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.

Proof. A morphism from a colimit into an object is the same as a compatible system of morphisms from the constituents of the limit into the object, see Remark 4.14.4. So

\begin{matrix} \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(\mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}} M_ i), Y) & = & \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(\mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}} M_ i, v(Y)) \\ & = & \mathop{\mathrm{lim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, v(Y)) \\ & = & \mathop{\mathrm{lim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(u(M_ i), Y) \end{matrix}

proves that u(\mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}} M_ i) is the colimit we are looking for. A similar argument works for the other statement. \square


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