Lemma 4.17.4. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of categories. Assume $\mathcal{I}$ is initial in $\mathcal{J}$. Then for every diagram $M : \mathcal{J} \to \mathcal{C}$ we have a canonical isomorphism

\[ \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \circ H = \mathop{\mathrm{lim}}\nolimits _\mathcal {J} M \]

if either side exists.

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