Lemma 14.21.11. Let \mathcal{C} be a category with finite coproducts and finite limits. Let V be a simplicial object of \mathcal{C}. In this case
Proof. By Lemma 14.13.4 the object on the left represents the functor which assigns to X the first set of the following equalities
The object on the right in the formula of the lemma is represented by the functor which assigns to X the last set in the sequence of equalities. This proves the result.
In the sequence of equalities we have used that \text{sk}_ n (X \times \Delta [n + 1]) = X \times \text{sk}_ n \Delta [n + 1] and that i_{n!}(X \times \text{sk}_ n \Delta [n + 1]) = X \times i_{n !} \text{sk}_ n \Delta [n + 1]. The first equality is obvious. For any (possibly truncated) simplicial object W of \mathcal{C} and any object X of \mathcal{C} denote temporarily \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) the (possibly truncated) simplicial set [n] \mapsto \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W_ n). From the definitions it follows that \mathop{\mathrm{Mor}}\nolimits (U \times X, W) = \mathop{\mathrm{Mor}}\nolimits (U, \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W)) for any (possibly truncated) simplicial set U. Hence
This proves the second equality used, and ends the proof of the lemma. \square
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