Lemma 14.13.4. With X and k as above. For any simplicial object V of \mathcal{C} we have the following canonical bijection
\mathop{\mathrm{Mor}}\nolimits _{\text{Simp}(\mathcal{C})}(X \times \Delta [k], V) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, V_ k).
which maps \gamma to the restriction of the morphism \gamma _ k to the component corresponding to \text{id}_{[k]}. Similarly, for any n \geq k, if W is an n-truncated simplicial object of \mathcal{C}, then we have
\mathop{\mathrm{Mor}}\nolimits _{\text{Simp}_ n(\mathcal{C})}(\text{sk}_ n(X \times \Delta [k]), W) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W_ k).
Proof.
A morphism \gamma : X \times \Delta [k] \to V is given by a family of morphisms \gamma _\alpha : X \to V_ n where \alpha : [n] \to [k]. The morphisms have to satisfy the rules that for all \varphi : [m] \to [n] the diagrams
\xymatrix{ X \ar[r]^{\gamma _\alpha } \ar[d]^{\text{id}_ X} & V_ n \ar[d]^{V(\varphi )} \\ X \ar[r]^{\gamma _{\alpha \circ \varphi }} & V_ m }
commute. Taking \alpha = \text{id}_{[k]}, we see that for any \varphi : [m] \to [k] we have \gamma _\varphi = V(\varphi ) \circ \gamma _{\text{id}_{[k]}}. Thus the morphism \gamma is determined by the value of \gamma on the component corresponding to \text{id}_{[k]}. Conversely, given such a morphism f : X \to V_ k we easily construct a morphism \gamma by putting \gamma _\alpha = V(\alpha ) \circ f.
The truncated case is similar, and left to the reader.
\square
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