Definition 14.14.1. Let $\mathcal{C}$ be a category with finite products. Let $V$ be a cosimplicial object of $\mathcal{C}$. Let $U$ be a simplicial set such that each $U_ n$ is finite nonempty. We define *$\mathop{\mathrm{Hom}}\nolimits (U, V)$* to be the cosimplicial object of $\mathcal{C}$ defined as follows:

we set $\mathop{\mathrm{Hom}}\nolimits (U, V)_ n = \prod _{u \in U_ n} V_ n$, in other words the unique object of $\mathcal{C}$ such that its $X$-valued points satisfy

\[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, \mathop{\mathrm{Hom}}\nolimits (U, V)_ n) = \text{Map}(U_ n, \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, V_ n)) \]and

for $\varphi : [m] \to [n]$ we take the map $\mathop{\mathrm{Hom}}\nolimits (U, V)_ m \to \mathop{\mathrm{Hom}}\nolimits (U, V)_ n$ given by $f \mapsto V(\varphi ) \circ f \circ U(\varphi )$ on $X$-valued points as above.

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