Dual version of Adjoint Functor theorems

Question

I am trying to dualize three versions of the adjoint functor theorem.

1. If $C$ and $D$ are locally small, $C$ is total (meaning the yoneda functor has left adjoint) then $F:C\rightarrow D$ has a right adjoint iff $F$ preserves colimits.
2. If $C$ is locally presentable and $D$ is locally small then $F:C\rightarrow D$ has a right adjoint iff $F$ preserves colimits.
3. If $C$ and $D$ are locally small and $C$ has small colimits, then $F$ has right adjoint iff $F$ preserves colimits and $(F\Rightarrow x)$ satisfies the solution set condition for all $x\in D$.

Now, in order to dualize the results I know I must replace:

$F$ has right adjoint iff $F$ preserves colimits

And instead have:

$G:D\rightarrow C$ has left adjoint iff $G$ preserves colimits.

But I am confused about what to for the conditions on the categories and for the dual of the 'solution set condition' property.

For the size conditions, must I swap the categories $C$ and $D$? For example:

1. $C,D$ locally small and $D$ is total
2. $C$ locally small and $D$ locally presentable
3. $C,D$ locally small, $D$ has all small limits

I think the last condition should be changed to $(x\rightarrow G)$ satisfies a dual version of the SSC.

Could someone please list the dual version of the three adjoint functor theorems I stated above?

Answer 1

The goal is to take the opposite of every category and functor in sight. If $F:C\to D$, then $F^{\mathrm{op}}:C^{\mathrm{op}}\to D^{\mathrm{op}}$. So for instance in 1, if $C^{\mathrm{op}}$ and $D^{\mathrm{op}}$ are locally small and $C^{\mathrm{op}}$ is total, then $F^{\mathrm{op}}$ has a right adjoint iff $F^{\mathrm{op}}$ preserves colimits. Now $C^{\mathrm{op}}$ is locally small if and only if $C$ is; $C^{\mathrm{op}}$ is total if and only if the Yoneda embedding $y:C^{\mathrm{op}}\to [C,\mathrm{Set}]$ has a left adjoint, if and only if the co-Yoneda embedding $y^{\mathrm{op}}:C\to [C,\mathrm{Set}]^{\mathrm{op}}$ has a right adjoint. In this case $C$ is called cototal. Finally, $F^{\mathrm{op}}$ has a right adjoint if and only if $F$ has a left adjoint and $F^{\mathrm{op}}$ preserves colimits if and only if $F$ preserves limits.

So, in the end, we see: a functor from a cototal locally small category to a locally small category has a left adjoint if and only if it preserves limits. You just have to be careful that cototal does not mean that the ordinary Yoneda embedding has a right adjoint! To remember this it helps that $C\to [C,\mathrm{Set}]^{\mathrm{op}}$ is the free completion of $C$ just as $C\to [C^{\mathrm{op}},\mathrm{Set}]$ is the free cocompletion.

The others are similar. It's your choice how much to flesh out the condition that $C^{\mathrm{op}}$ be locally presentable, but we get the same conclusion as above. Perhaps the trickiest dualization remaining is in $3$, but again one just has to carefully dualize each category and functor. We need $F^{\mathrm{op}}\Rightarrow x$ to satisfy the solution set condition for each $x\in D^{\mathrm{op}}$. $F^{\mathrm{op}}\Rightarrow x$ is the category of morphisms $F^{\mathrm{op}}(y)\to x$ in $D^{\mathrm{op}}$, so is isomorphic to the opposite category of morphisms $x\to F(y)$ in $D$, that is, $(F^{\mathrm{op}}\Rightarrow x)=(x\Rightarrow F)^{\mathrm{op}}$. And for the latter to satisfy the solution set condition means that there exists a set of maps $x\to F(y_i)$ such that every map $x\to F(y)$ factors through via a map $F(y)\to F(y_i)$ in $D$ (not $D^{\mathrm{op}}$.) As one should predict, this is a weak form of asking that $x\Rightarrow F$ have a terminal object.