Lemma 4.24.5 (0038)—The Stacks project

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.

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. $\square$

The Stacks project

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