ScalingStacks

Proof. Since the cells are contained in $\Upsilon_{n}$, it follows that $j_{!}j^{*}C_{i}\simeq C_{i}$. Suppose that $Y\in\precat_{(\infty,n)}$; then the unit $Y\to j_{\ast}j^{\ast}Y$ induces an equivalence

$$\map(C_{i},Y)\simeq\map(j_{!}j^{*}C_{i},Y)\simeq\map(C_{i},j_{\ast}j^{\ast}Y)$$

for any cell $C_{i}$.

Now consider the smallest subcategory of $\cat_{(\infty,n)}$ consisting of objects $X$ such that the unit map induces an equivalence $\map(X,Y)\simeq\map(X,j_{\ast}j^{\ast}Y)$ for all $Y\in\precat_{(\infty,n)}$. As we have seen this subcategory contains the cells. It is also closed under colimits since if we write $X\simeq\colim_{\alpha}X_{\alpha}$, where all the $X_{\alpha}$ are in this subcategory, then

$$\map(X,Y)\simeq\lim_{\alpha}\map(X_{\alpha},Y)\simeq\lim_{\alpha}\map(X_{\alpha},j_{\ast}j^{\ast}Y)\simeq\map(X,j_{\ast}j^{\ast}Y).$$

It follows that this subcategory is all of $\cat_{(\infty,n)}$, and thus $j^{\ast}$ induces an equivalence $\map(X,Y)\simeq\map(j^{\ast}X,j^{\ast}Y)$. ∎