For a diagram category $\Gamma$ and and a cocomplete category $\mathcal{C}$, we have an equivalence $$\mathrm{Fun}(\Gamma,\mathcal{C}) \simeq \mathrm{Adj}(Set^{\Gamma^{Op}},C)$$ where for $F: \Gamma \to C$ we have a pair of adjoint functors $\mathcal{L}_F: Set^{\Gamma^{Op}} \to \mathcal{C}: S \mapsto \mathrm{colim}_{h_\gamma \to S}F(\gamma)$ (left adjoint) and $\mathcal{N}_F: \mathcal{C} \to Set^{\Gamma^{Op}}: c \mapsto \mathcal{C}(F(-),c)$ (right adjoint).

For $\Gamma = \Delta$, this states that a cosimplicial object in $\mathcal{C}$ defines an adjunction between simplicial sets and $\mathcal{C}$. For example, when $\mathcal{C}=Top$ and the cosimpicial object consists of standard toplogical simplices, $\mathcal{L}_F$ is geometric realization and $\mathcal{N}_F$ is singular set.

Generally for $\Gamma=\Delta$, the explicit formula for $\mathcal{L}_F$ is $$\mathcal{L}_F(S_\bullet) = Coeq(\amalg_{\phi:[m]\to[n]}F([m])\times S_n \rightrightarrows \amalg_{[n]}F([n])\times S_n).$$

When instead of a simplicial set $S_\bullet$ we have a cosimplicial category $A^\bullet$, there exists a (somewhat dual) notion of totalization: $$\mathrm{Tot}(A^\bullet):=Eq(\Pi_{[n]}\mathrm{Fun}(Iso(n),A^n) \rightrightarrows \Pi_{[n] \to [m]}\mathrm{Fun}(Iso(n),A^m))$$

where $Iso(n)$ stands for the category "string of $n$ isomorphisms". This totalization is a correct notion (e.g. it works well for $A^\bullet$ being sheaves on a Čech nerve of a cover).

I would expect it to be a part of some general picture similar to $\mathrm{Fun}(\Gamma,\mathcal{C}) \simeq \mathrm{Adj}(Set^{\Gamma^{Op}},C)$, only in enriched setting (sets are replaced by categories) and for $\Gamma=\Delta^{Op}$. However, it is not exactly the picture above, as $Iso(n)$ is a functor $\Delta \to Cat$ not $\Delta^{Op} \to Cat$.

What is this general picture for totalizations? Is there an adjoint functor?