Lemma 14.19.6. Let n be an integer \geq 1. Let U be a n-truncated simplicial object of \mathcal{C}. Consider the contravariant functor from \mathcal{C} to \textit{Sets} which associates to an object T the set
\{ (f_0, \ldots , f_{n + 1}) \in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(T, U_ n) \mid d^ n_{j - 1} \circ f_ i = d^ n_ i \circ f_ j \ \forall \ 0\leq i < j\leq n + 1\}
If this functor is representable by some object U_{n + 1} of \mathcal{C}, then there exists an (n + 1)-truncated simplicial object \tilde U, with \text{sk}_ n \tilde U = U and \tilde U_{n + 1} = U_{n + 1} such that the following adjointness holds
\mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_{n + 1}(\mathcal{C})}(V, \tilde U) = \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ n(\mathcal{C})}(\text{sk}_ nV, U)
Proof.
By Lemma 14.19.3 there are identifications
U_ i = \mathop{\mathrm{lim}}\nolimits _{(\Delta /[i])_{\leq n}^{opp}} U(i)
for 0 \leq i \leq n. By Lemma 14.19.5 we have
U_{n + 1} = \mathop{\mathrm{lim}}\nolimits _{(\Delta /[n + 1])_{\leq n}^{opp}} U(n).
Thus we may define for any \varphi : [i] \to [j] with i, j \leq n + 1 the corresponding map \tilde U(\varphi ) : \tilde U_ j \to \tilde U_ i exactly as in Lemma 14.19.2. This defines an (n + 1)-truncated simplicial object \tilde U with \text{sk}_ n \tilde U = U.
To see the adjointness we argue as follows. Given any element \gamma : \text{sk}_ n V \to U of the right hand side of the formula consider the morphisms f_ i = \gamma _ n \circ d^{n + 1}_ i : V_{n + 1} \to V_ n \to U_ n. These clearly satisfy the relations d^ n_{j - 1} \circ f_ i = d^ n_ i \circ f_ j and hence define a unique morphism V_{n + 1} \to U_{n + 1} by our choice of U_{n + 1}. Conversely, given a morphism \gamma ' : V \to \tilde U of the left hand side we can simply restrict to \Delta _{\leq n} to get an element of the right hand side. We leave it to the reader to show these are mutually inverse constructions.
\square
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