Category which is not a subcategory of a complete category By Yoneda lemma every small category $\mathcal{C}$ can be embedded in the cocomplete (and complete) category $[C^{op} ,\textbf{Set}]$. Most examples I know of large categories which are not complete, like the category of fields, are subcategories of a complete and cocomplete category. Is there an example of a large category which is not contained in a complete (cocomplete) category.
Note: When I say category I mean a locally small category.
 A: $\DeclareMathOperator{\colim}{colim}\newcommand{\cat}{\mathbf}\DeclareMathOperator{\Hom}{Hom}$ For any locally small category $\mathbf C$, you can construct its cocompletion $\hat{\mathbf C}$ as the following locally small category:


*

*The class of objects is the class of small diagrams in $\mathbf C$ (i.e. functors from a small category to a small subcategory of $\mathbf C$)

*For any two small diagrams $\cat D\xrightarrow{F}\cat C\xleftarrow{G}\cat E$,  the set of morphisms $\colim F\Rightarrow\colim G$ consists of morphisms $FA\xrightarrow{\eta_A} GX_A$ in $\cat C$, indexed by $A\in\cat D$, up to the equivalence relation that two families $FA\xrightarrow{\eta_A}GX_A$ and $FA\xrightarrow{\epsilon_A}GY_A$ are equivalent if $FA\xrightarrow{\epsilon_A}GY_A$ factors as $FA\xrightarrow{\eta_A}GX_A\dashrightarrow GY_A$ for some $X_A\dashrightarrow Y_A$. This is something like families of compatible germs of morphisms from the diagram $F$ to the diagram $G$.


One reason to construct it this way because when $\cat C$ is large, it is handwavy to say "the subcategory of small colimits of representable contravariant functors in $[\cat C^{op},\cat{Set}]$" since the latter is not a category (lacks a class of objects!). But if $\cat C$ is small, then $\hat{\cat C}=[\cat C^{op},\cat{Set}]$. Indeed ( once we identify $\hat{\cat C}$ with the small colimits of representable functors) this follows from the fact that every $\cat C^{op}\to\cat{Set}$ is the colimit of the its diagram of elements obtained by applying the obvious forgetful functor to the category whose objects are pairs $X\in\cat C$ and $x\in FX$, and morphisms $(X,x)\to(Y,Y)$ are $X\to Y\in\cat C$ so that $Ff(x)=y\in Y$.
To see that $\hat{\cat C}$ is the category of small colimits of representable functors, proceed as follows. First, identify $\cat C$ with the class of singleton diagrams; evidently $\cat C$ embeds in $\hat{\cat C}$ by definition.
Furthermore, $A\Rightarrow\colim G$ is exactly a morphism $A\to GY$ for some $Y\in\cat E$ up to the relation that $A\to GY\to GZ$ are considered equivalent for every $Y\to Z\in\cat E$. This is exactly the statement that $A\Rightarrow G$ is an element of $\colim_{Y\in\cat E}\Hom_{\cat C}(A,GY)$. In particular, $\cat C\hookrightarrow\hat{\cat C}$ is full and faithful. By pointwise computation of colimits, it follows that $\Hom_{\hat{\cat C}}(A,G)=\colim_{Y\in\mathbf E}\Hom_{\hat{\cat C}}(A,GY)=\Hom_{\hat{\cat C}}(A,\colim_{Y\in\mathbf E}GY)$, and hence by the Yoneda lemma that $G=\colim_{Y\in\mathbf E}GY$. 
Furthermore, by pointwise computation of colimits and the Yoneda lemma, $\hat{\cat C}$ is cocomplete because iterated small colimits are still small. Finally, the Yoneda embedding $\cat C\hookrightarrow\hat{\cat C}$ is continuous by the pointwise computation of limits. But warning: it is not cocontinuous: colimits are freely added much the same way free variables are added.
Dualizing, you can also get the completion of $\cat C$. 
However, note that $\hat{\cat C}$ does not have to be complete, e.g. it lacks a terminal object when $\cat C$ is large and discrete. According to some notes by Todd Trimble, it seems the embedding of $\cat C$ into the cocompletion of the completion will be an embedding into a complete cocomplete locally small category.
