## When are terminal “models” of a category guaranteed to exist?

[The following is a "tightening-up" of a question I asked earlier today to which I received no answer.]

Let ${\bf Cat}$ denote the category of small categories. Recall that for a category $\mathcal{C}$ and a functor $F\colon\mathcal{C}\to{\bf Cat}$, the Grothendieck construction of $F$, which I'll denote $\int F$, is a category and it comes equipped with a natural fibration $p_F\colon\int F\to\mathcal{C}$.

[For reference: The objects of $\int F$ are pairs $(c,x)$ where $c\in{\bf Ob}(\mathcal{C})$ and $x\in{\bf Ob}(F(\mathcal{C}))$, and a morphism $(c,x)\to(c',x')$ is a pair $(f,g)$ where $f\colon c\to c'$ in $\mathcal{C}$ and $g\colon F(f)(x)\to x'$ in the category $F(c')$.]

Now, given a category ${\mathcal D}$, I'll define a model of ${\mathcal D}$ to be a pair $({\mathcal C},F,e)$ where $\mathcal{C}$ is a category, $F\colon \mathcal{C}\to{\bf Cat}$ is a functor, and $e\colon\int F\to\mathcal{D}$ is a natural isomorphism. I will sometimes leave out $e$ if it is obvious or annoying. A morphism of models $(\mathcal{C},F)\to(\mathcal{C}',F')$ is a functor $a\colon{\mathcal C}\to{\mathcal C'}$ such that $a\circ p_F\cong p_{F'}$. Denote the category of models of $\mathcal{D}$ by $\mathcal{D}-{\bf Model}$. [Feel free to fix up this definition if it is not quite "the good one".]

Let ${\bf S}\subseteq{\bf Cat}$ denote a subcategory. Say that a model $(\mathcal{C},F)$ of ${\mathcal D}$ is ${\bf S}$-special if $F\colon\mathcal{C}\to{\bf Cat}$ factors through ${\bf S}$. The ${\bf S}$-special models of $\mathcal{D}$ constitute a full subcategory of $\mathcal{D}-{\bf Model}$.

For some subcategories ${\bf S}$, the category of ${\bf S}$-special models of $\mathcal{D}$ will have a terminal object. For example, in case ${\bf S}={\bf Cat}$ is everything or (trivially) in case ${\bf S}=$ {$\ast$} consists only of the terminal category.

But what if ${\bf S}={\bf Set}$ or ${\bf S}={\bf Poset}$? What if ${\bf S}={\bf Monoid}$? Is a terminal ${\bf S}$-special model guaranteed?

Question: In what case of ${\bf S}\subset{\bf Cat}$ is the category of ${\bf S}$-special models of $\mathcal{D}$ guaranteed to have terminal object? What if we restrict $\mathcal{D}$ in some way?

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