Hello,


I with to consider the following statement:


If $C$ is a cocomplete category having a dense small full subcategory $D$, then $C$ is complete.


(a full subcategory $D$ is dense in $C$ if every element of $C$ is canonical colimit of elements of $D$...)


I think I know how to prove it (I give proof below), and I want someone to reassure me that this statement is true exactly as stated, as it seems a little bit surprising.


Proof sketch:


Consider the functor $Y : C \to psh(D)$, where $psh(D)$ denotes the category of presheaves on $D$ (Yoneda functor, i.e. $Y(X)(A) = Hom (A, X)$). $D$ being dense in $C$ is equivalent to this functor being fully faithful. Futhermore, we have a functor $F: psh(D) \to C$, namely, the one which extends the inclusion $D \to C$ by cocontinuity (as presheaf categories have the property of being free cocompletions: <http://ncatlab.org/nlab/show/free+cocompletion>). Then one can see that $Y$ is right adjoint to $F$. So this renders $C$ as reflective subcategory of $psh(D)$ ( <http://ncatlab.org/nlab/show/reflective+subcategory>). Now, $psh(D)$ is complete, and so every reflective subcategory of it. hence, $C$ is complete.


Thank you,
Sasha



Any cocomplete category with a dense small full subcategory is complete?
This is a well-known theorem, you can find it for example in [Abstract and concrete categories - the Joy of Cats](http://katmat.math.uni-bremen.de/acc/acc.pdf), Theorem 12.12. The proof there uses that a cocomplete category with a weakly terminal object has a terminal object (the preparation for the Freyd's adjoint functor theorem).


