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# CocompleteAnycocomplete category (s.t....)withadensesmallfullsubcategory is complete?

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:

Thank you, Sasha