$\newcommand\Sketch{\mathit{Sketch}}\newcommand\Set{\mathit{Set}} \DeclareMathOperator\Lim{Lim}\DeclareMathOperator\Colim{Colim}\DeclareMathOperator\Mod{Mod}\newcommand\mod{\operatorname{mod}}\DeclareMathOperator\Ind{Ind}\DeclareMathOperator\Fun{Fun} $Recall that a sketch $s = (\mathcal A_s, \mathcal L_s, \mathcal C_s)$ on a locally small category $\mathcal A_s$ comprises a collection $\mathcal L_s$ of small cones and a collection $\mathcal C_s$ of small cocones. A morphism $s \to t$ is a functor $A_s \to A_t$ carrying $\mathcal L_s$ to $\mathcal L_t$ and $\mathcal C_s$ to $\mathcal C_t$. We have a category, even a 2-category, $\Sketch$.
(I really do want to consider (limit, colimit) sketches and not just limit sketches here -- see the last bullet point below for an example of an interesting phenomenon which requires colimits in the sketches to see.)
Even better — there is a canonical symmetric monoidal closed structure on $\Sketch$ (so that $\Sketch$ is self-enriched). The unit is $i := (\ast, \emptyset, \emptyset)$. The tensor $s \otimes t$ is the coarsest sketch on $\mathcal A_s \times \mathcal A_t$ such that the functors $S \times (-) : \mathcal A_t \to \mathcal A_s \times \mathcal A_t \leftarrow \mathcal A_s : (-) \times T$ are sketch morphisms for $S \in \mathcal A_s, T \in \mathcal A_t$. The internal hom $\langle s, t \rangle$ is the sketch on $\Sketch(s,t)$ where $\lambda \in \mathcal L_{\langle s, t \rangle}$ iff $\lambda(-)(S) \in \mathcal L_t$ for any $S \in s$; $\mathcal C_{\langle s, t \rangle}$ is defined similarly.
A model of a sketch $s$ is a morphism $s \to d$ where $d:= (\Set, \Lim, \Colim)$. Write $\Mod(s) = \Sketch(s, d)$ for the category of models, and $\mod(s) = \langle s, d \rangle$ for the sketch of models. For any sketch $s$, there is a canonical sketch map $s \to \mod(\mod(s))$.
Question 1: When is the canonical map $s \to \mod(\mod(s))$ an equivalence?
Question 2: Is the canonical map $\mod(s) \to \mod(\mod(\mod(s)))$ always an equivalence?
Question 3: How do (1) and (2) look if we make everything in sight enriched in some nice symmetric monoidal category $\mathcal V$?
For instance, $\mod(\mod(i))$ differs from $i$ only by adjoining absolute limits and colimits (it has an equivalent underlying category). We have $\mod(i) = \mod(\mod(\mod(i))) = d$, and $(d, \mod(\mod(i))$ are a dual pair under $\mod$.
For instance, if $s = (C, \mathrm{finitelimit}, \emptyset)$ for some small, finitely-complete category $C$, then $\Mod(s) = (\Ind(C^\text{op}), \mathrm{limit}, \mathrm{filteredcolimit})$, and $\Mod(\Mod(s))$ differs from $s$ only by the addition of all absolute limits and colimits to the sketch. In particular, in this case we have $\mod(s) \simeq \mod(\mod(\mod(s)))$. Thus we basically recover Gabriel–Ulmer duality.
If $s = (C, \emptyset, \emptyset)$ for some small category $C$, then $\mod(S) = (\Fun(C,\Set), \mathrm{limit}, \mathrm{colimit})$, and $\mod(\mod(s))$ is the idempotent completion of $C$, with just the absolute limits and colimits sketched, and again we have $\mod(s) \simeq \mod(\mod(\mod(s)))$.
If we make everything additive, I believe that the sketch for a chain complex and the sketch for an exact chain complex are dual to each other under $\mod$ (the exact chain complex sketch is not just a limit sketch; it has colimits in it as well).
Question 4: Are there other interesting examples of dual pairs $s \simeq \mod(t)$, $t \simeq \mod(s)$?
Question 5: Is it always the case that $\mod(\mod(s))$ is obtained from $s$ by passing to the idempotent completion and adjoining absolute limits and colimits to the sketch, or can more interesting things happen as well?
