The Stacks project

Remark 4.22.7. Let $\mathcal{C}$ be a category. Let $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ be cofiltered diagrams in $\mathcal{C}$. Consider the functors $A, B : \mathcal{C} \to \textit{Sets}$ defined by

\[ A(X) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(F(i), X) \quad \text{and}\quad B(X) = \mathop{\mathrm{colim}}\nolimits _ j \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(G(j), X) \]

We claim that a morphism of pro-systems from $F$ to $G$ is the same thing as a transformation of functors $t : B \to A$. Namely, given $t$ we can apply $t$ to the class of $\text{id}_{G(j)}$ in $B(G(j))$ to get a compatible system of elements $\xi _ j \in A(G(j)) = \mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(F(i), G(j))$ which is exactly our definition of a morphism in $\text{Pro-}\mathcal{C}$ in Remark 4.22.5. We omit the construction of a transformation $B \to A$ given a morphism of pro-objects from $F$ to $G$.

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