Lemma 14.15.2. With $X$, $k$ and $U$ as above.

For any simplicial object $V$ of $\mathcal{C}$ we have the following canonical bijection

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(V, U) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V_ k, X). \]which maps $\gamma $ to the morphism $\gamma _ k$ composed with the projection onto the factor corresponding to $\text{id}_{[k]}$.

Similarly, if $W$ is an $k$-truncated simplicial object of $\mathcal{C}$, then we have

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ k(\mathcal{C})}(W, \text{sk}_ k U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(W_ k, X). \]The object $U$ constructed above is an incarnation of $\mathop{\mathrm{Hom}}\nolimits (C[k], X)$ where $C[k]$ is the cosimplicial set from Example 14.5.6.

## Comments (2)

Comment #1299 by JuanPablo on

Comment #1309 by Johan on