Consider a simplicial category $\cal D$, and a cosimplicial object $c\colon \Delta\to \cal D$. Let $y\colon \Delta\to \bf sSet$ denote the Yoneda embedding.
How do you characterize the limit of $c$ weighted by $y$ (provided it exists)?
Partial answer: Whatever it is such an object,its associated representable presheaf must behave in such a way that $$ {\cal D}\big(d,\textstyle \text{lim}^yc\big)\cong {\bf Set}^{\Delta^\text{op}\times\Delta}\big( y,{\cal D}(d,c-) \big) $$ or in other words $$\begin{align*} {\cal D}\big(d, \text{lim}^yc\big) &\cong {\bf Set}^{\Delta^\text{op}\times\Delta}\big( y,{\cal D}(d,c-) \big)\\ & \int_{n\in\Delta}{\bf sSet}\big( \Delta_n, {\cal D}(d, c_n) \big)\\ &\cong \int_{n\in\Delta} {\cal D}(d, c_n)_n \end{align*}$$ which is the "cotrace" (?) of the profunctor $\Delta^\text{op}\times\Delta\to{\bf Sets}\colon (m,n)\mapsto {\cal D}(d, c_n)_m$. Is this something meaningful? How can it be generalized to the case of a functor $F\colon \cal C\to D$, where $\cal D$ is a $\widehat{\cal C}$-category?