Lemma 114.10.3. 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).