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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?

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There are a number of ways of expressing this duality.

  • The colimit in the question may be written as the coend $\mathcal L_F(S_\bullet) = \int^{[n] \in \Delta} F([n]) \times S_n$, while the limit may be written as $\mathrm{Tot}(A^\bullet) = \int_{[n] \in \Delta} \mathrm{Hom}(Iso([n]), A^n))$.

  • The colimit in the question is the weighted colimit $\mathcal L_F(S_\bullet) = F \otimes S$ (the colimit "of" the functor $S$, "weighted" by the presheaf $F$). The limit in the question is the weighted limit $\mathrm{Tot}(A^\bullet) = \{Iso,A\}$ (the limit "of" the functor $A$, "weighted" by the copresheaf $A$).

  • The colimit in the question computes the the left Kan extension $\mathcal L_{(-)}(S_\bullet)$ of the functor $S_\bullet$ along the Yoneda embedding $\Delta \to Fun(\Delta^{op},\mathsf{Set})$. The limit in the question (generalized from the case of the functor $Iso$ in the obvious way) computes the right Kan extension $\mathrm{Tot}^{(-)}(A^\bullet)$ of the functor $A^\bullet$ along the coYoneda embedding $\Delta \to Fun(\Delta,\mathsf{Set})^{op}$.

  • The category of presheaves $Fun(\Delta^{op},\mathsf{Set})$ is the free cocompletion of $\Delta$, so that $Fun(\Delta \simeq Fun^{cocts}(Set^{\Delta^{op}},\mathcal C)$ for cocomplete $\mathcal C$ (where $Fun^{cocts}$ means cocontinuous functors). Dually, the opposite of the category copresheaves $Fun(\Delta,\mathsf{Set})^{op}$ is the free completion of $\Delta$, i.e. there is an equivalence $Fun(\Delta,\mathcal{C}) \simeq Fun^{cts}((Set^{\Delta})^{op},\mathcal{C})$ (where $Fun^{cts}$ means continuous functors).

  • Applying the adjoint functor theorem to the previous point, we obtain the equivalence mentioned in the question $Fun(\Delta,\mathcal C) \simeq LAdj(Set^{\Delta^{op}},\mathcal C)$ (the cateogry of left adjoint functors). Dually, we have $Fun(\Delta,\mathcal{C}) \simeq RAdj((Set^{\Delta})^{op},\mathcal{C})$ (the category of right adjoint functors.

And all of this works for any small $\Gamma$ in place of $\Delta$, and any complete and cocomplete $\mathcal{C}$.

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