I am reading the paper <http://www.numdam.org/article/CTGDC_2001__42_1_51_0.pdf> fixing the implication $(ii)\Rightarrow (i)$ of Theorem 1.39 of Adamek-Rosicky's book. The correct statement is: if $\mathcal{K}$ is a reflective subcategory of a LFP category $\mathcal{L}$ closed under filtered colimits and such that the theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is a quotient, then $\mathcal{K}$ is a small $\omega$-orthogonality class of $\mathcal{L}$.



> 
> I can "prove" that the theory of the embedding
>  $\mathcal{K}\subset\mathcal{L}$ is always a quotient, which is
>  obviously wrong. Where am I wrong in what follows ?
> 
> 
> 


Let $r:\mathcal{L} \to \mathcal{K}$ be the left adjoint of the inclusion functor $i:\mathcal{K}\subset \mathcal{L}$. I am using the terminology of <https://ncatlab.org/nlab/show/Gabriel-Ulmer+duality>: $\text{Lex}$ is the category of finitely complete (essentially) small categories together with the finite limit-preserving functors. The theory of the embedding $\mathcal{K}\subset \mathcal{L}$ is the $\text{Lex}$-functor $r^{op}:\mathcal{L}\_{fp}^{op} \to \mathcal{K}\_{fp}^{op}$. To prove that it is a quotient, we have to prove two things: 1) that every object $B$ of $\mathcal{K}\_{fp}^{op}$ is isomorphic to an object $r^{op}A$: indeed, $r^{op}i^{op}B\cong B$ since $\mathcal{K}$ is a full subcategory of $\mathcal{L}$; 2) that every map $b:X\to r^{op}A$ factors as a composite $X\to r^{op}A' \to r^{op}A$ where the left-hand map is an isomorphism and the right-hand map is $r^{op}f$ for some $f:A'\to A$, which seems to be a consequence of
$\mathcal{K}^{op}\_{fp}(X,r^{op}A) \cong \mathcal{K}(rA,X) \cong \mathcal{L}(A,A') \cong \mathcal{L}^{op}\_{fp}(A',A)$ with $A'=iX$. 



About small $\omega$-orthogonality classes and Gabriel-Ulmer duality
I understand where is the mistake. $r$ does preserve finite presentability (the proof is straightforward and it is due to the fact that it is a left adjoint of a functor preserving filtered colimits). But there is no reason for $A'=iX$ to be finitely presentable.


