The Stacks project

Lemma 4.14.8. Suppose that $M : \mathcal{I} \to \mathcal{C}$, and $N : \mathcal{J} \to \mathcal{C}$ are diagrams whose colimits exist. Suppose $H : \mathcal{I} \to \mathcal{J}$ is a functor, and suppose $t : M \to N \circ H$ is a transformation of functors. Then there is a unique morphism

\[ \theta : \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \longrightarrow \mathop{\mathrm{colim}}\nolimits _\mathcal {J} N \]

such that all the diagrams

\[ \xymatrix{ M_ i \ar[d]_{t_ i} \ar[r] & \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \ar[d]^{\theta } \\ N_{H(i)} \ar[r] & \mathop{\mathrm{colim}}\nolimits _\mathcal {J} N } \]


Proof. Omitted. $\square$

Comments (4)

Comment #76 by Keenan Kidwell on

I think the in the diagram should be .

Comment #83 by on

Well, I am following the notation: If is a transformation of functors then is made up of maps where varies over objects of the source category of and . Since in the statement of the lemma and have source , I think the notation makes sense. I am not saying the notation is perfect though...

Comment #88 by Keenan Kidwell on

Both and have source category , right? So wouldn't writing be in accordance with the notation?

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  • 7 comment(s) on Section 4.14: Limits and colimits

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