In _Coalgebras in symmetric monoidal categories of spectra_, Péroux and Shipley prove the following (Lemma 2.4):

> Let $\mathcal{C}=\mathsf{Sets},\mathsf{Top}$, or $\mathsf{sSets}$. The free basepoint functor
> $$(-)^+\colon(\mathcal{C},\times,\mathrm{pt})\to(\mathcal{C}_*,\wedge,S^0)$$
> lifts to an equivalence of categories
> $$(-)^+\colon\mathsf{CoMon}(\mathcal{C})\dashrightarrow\mathsf{CoMon}(\mathcal{C}_*).$$
> As such, any comonoid in $\mathcal{C}_*$ is isomorphic to one of the form $(C^+,\Delta^+_{C},\epsilon^+_C)$ with $(C,\Delta_{C},\epsilon_C)$ a comonoid in $\mathcal{C}$.

 - Is this result valid in general for $\mathcal{C}$ a bicomplete Cartesian closed symmetric monoidal category, where $(\mathcal{C}_*,\wedge,S^0)$ is now the symmetric monoidal category in [Riehl, _Categorical Homotopy Theory_, Construction 3.3.14](https://math.jhu.edu/~eriehl/cathtpy.pdf#thm.3.3.14)?
 - Let $\mathcal{C}$ be a bicomplete Cartesian closed symmetric monoidal $\infty$-category.
     1. Can we similarly build a symmetric monoidal $\infty$-category structure  $(\wedge,S^0)$ on the category $\mathcal{C}_*$ of pointed objects of $\mathcal{C}$ (i.e. the coslice $\infty$-category $\mathcal{C}_{*/}$)?
     2. If this is the case, do we have an analogue of Péroux–Shipley's result for $\mathbb{E}_{k}$-comonoids in $(\mathcal{C}_*,\wedge,S^0)$, where $1\leq k\leq\infty$?
     3. (If Item 1 fails, is Item 2 nevertheless true for $\mathcal{C}=\mathcal{S}$, the $\infty$-category of spaces?)