Lemma 4.14.7. 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

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

commute.

Proof. Omitted. $\square$

Comment #76 by Keenan Kidwell on

I think the $t_{H(i)}$ in the diagram should be $t_i$.

Comment #83 by on

Well, I am following the notation: If $t : F \to G$ is a transformation of functors then $t$ is made up of maps $t_X : F(X) \to G(X)$ where $X$ varies over objects of the source category of $F$ and $G$. Since in the statement of the lemma $M$ and $N \circ H$ have source $\mathcal{J}$, I think the notation makes sense. I am not saying the notation is perfect though...

Comment #88 by Keenan Kidwell on

Both $M$ and $N\circ H$ have source category $I$, right? So wouldn't writing $t_i$ be in accordance with the notation?

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

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