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What are the contributions of Beck and Chevalley respectively associated with the Beck-Chevalley condition


Consider a locally finitely presentable category C. Denote by Fin(C) the full subcategory of finitely presentable objects. Since C is uniquely determined by the subcategory Fin(C), the following question makes sense: 

What conditions on Fin(C) will ensure that C is regular? 

Is there any paper dealing with this kind of questions? 


In Categorical Geometry, Chapter 4, Section 4.3 I proved the following: 

If X is any object we denote by B(X) the poset of direct subobjects of X. The following Proposition 4.3.5 holds for any extensive category: 

Proposition 4.3.5. B(X) is a Boolean algebra. 

Proof. Suppose U and V are two direct subobjects of an object X. Then 

X = (U + Uc) Ç (V + Vc) = (U Ç V) + (Uc Ç V) + (U Ç Vc) + (Uc Ç Vc). 
(U Ç V) + (Uc Ç V) + (U Ç Vc
are direct subobjects. But 
U Ç V = U Ù V,
and we have the formula 
(U Ç V) + (Uc Ç V) + (U Ç Vc) = U Ú V
in B(X) as (U Ç V) + (U Ç Vc) = U  and  (U Ç V) + (Uc Ç V) = V. Thus B(X) is a lattice. 
If W is another direct subobject of X, then 
X = W Ç Uc + W Ç U + Wc 
implies that 
(W Ç U)c = W Ç Uc + Wc
(W Ç U)c Ç W = [(W Ç Uc) + Wc] Ç W = (W Ç Uc) Ç W.
(W Ç V)c = W Ç Vc Ç W.
We have 
    W Ç (U Ú V
= W Ç [(U Ç V) + (Uc Ç V) + (U Ç Vc)]  
= W Ç U Ç V + W Ç Uc Ç V + W Ç U Ç Vc  
= (W Ç U) Ç (W Ç V) + (W Ç Uc) Ç (W Ç V) + (W Ç U) Ç (W Ç Vc)  
= (W Ç U) Ç (W Ç V) + (W Ç U)c Ç (W Ç V) + (W Ç U) Ç (W Ç V)c  
= (W Ç U) Ú (W Ç V). 
This shows that B(X) is a distributive lattice. Clearly Uc is the complement of U in B(X). Thus B(X) is a Boolean algebra. n 

This has been proved by Diers in [Diers 1986, p.24, Proposition 1.3.3] in the dual situation for objects in a locally indecomposable category (= the dual of a coherent analytic category). Since this is a very fundamental fact, I would like to know whether it has already been covered in literature? 
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