Let $\Lambda$ and $C$ be categories, with $\Lambda$ small and $C$ complete. Let $F : \Lambda \rightarrow C$ be a diagram, where $C$ is cotensored over simplicial sets; there is then a functor $[\Delta^{op}, \text{Set}]^{op} \times C \rightarrow C$, written $(X, a) \mapsto a^X$. We can form the end $\int_{\lambda} F(\lambda)^{N(\lambda/\Lambda)}$.
How can we view $C = \text{Top}$ as having a map $[\Delta^{op}, \text{Set}]^{op} \times C \rightarrow C$? That is, what is the typical way to do this? I was thinking to use that $\text{Top}$ is cotensored over itself, and then to use the realization map.
How can we view $C = \text{Grpd}$, the category of small groupoids, as having a map $[\Delta^{op}, \text{Set}]^{op} \times C \rightarrow C$. I was thinking to use that $\text{Grpd}$ is cotensored over itself, and to use the realization map induced by the covariant functor $\Delta \rightarrow \text{Grpd}$ sending $[n]$ to the contractible groupoid with $n+1$ elements.
Under what conditions is this limit the same as the homotopy limit of $F$ in $C$? Does $C$ have to be a quillen category for this to be true?
In a simplicial model category, for instance, your formula always defines a right derived functor of limit, so agrees with the "global" definition of homotopy limit.