The Stacks project

Lemma 115.10.3. Let $\mathcal{C}$ be a category. Let $X$ be an object of $\mathcal{C}$ such that the self products $X \times \ldots \times X$ exist. Let $k \geq 0$ and let $C[k]$ be as in Simplicial, Example 14.5.6. With notation as in Simplicial, Lemma 14.15.2 the canonical map

\[ \mathop{\mathrm{Hom}}\nolimits (C[k], X)_1 \longrightarrow (\text{cosk}_0 \text{sk}_0 \mathop{\mathrm{Hom}}\nolimits (C[k], X))_1 \]

is identified with the map

\[ \prod \nolimits _{\alpha : [k] \to [1]} X \longrightarrow X \times X \]

which is the projection onto the factors where $\alpha $ is a constant map.

Proof. This is shown in the proof of Hypercoverings, Lemma 25.7.3. $\square$

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