Does the following characterize local presentability?

Question

Let $\mathcal C$ be a cocomplete category. Consider the following two conditions:

1. $\mathcal C$ is locally presentable.
2. The Yoneda embedding $$\mathcal C \hookrightarrow \{\text{continuous functors } \mathcal C^{\mathrm{op}} \to \mathrm{SET}\}$$ is an equivalence of categories. (By the Yoneda lemma, it suffices that it be essentially surjective.)

I know how to prove 1$\Rightarrow$2. Does the converse hold?

Answer 1

The following argument is from Martin Brandenburg's comment to the question linked to by Omar Antolin-Camarena. As you say, by the Yoneda lemma condition #2 holds iff every continuous functor $C^{op} \to \text{Set}$ is representable, or equivalently has a left adjoint. By the special adjoint functor theorem, it suffices that $C$ is cocomplete, co-well-powered, and has a small generating set.

$C = \text{Top}$ satisfies all of these conditions but is not locally presentable. Another example is $C = \text{Set}^{op}$, which is generated by $2$ but which cannot be locally presentable because it is the opposite of a nontrivial locally presentable category. (If $C$ and $C^{op}$ are both locally presentable then $C$ is a preorder.)