Lemma 4.14.9. Suppose that $M : \mathcal{I} \to \mathcal{C}$, and $N : \mathcal{J} \to \mathcal{C}$ are diagrams whose limits exist. Suppose $H : \mathcal{I} \to \mathcal{J}$ is a functor, and suppose $t : N \circ H \to M$ is a transformation of functors. Then there is a unique morphism

$\theta : \mathop{\mathrm{lim}}\nolimits _\mathcal {J} N \longrightarrow \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$

such that all the diagrams

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

commute.

Proof. Omitted. $\square$

Comment #1017 by correction_bot on

In the diagram, $t_{H(i)}$ should be $t_i$.

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