Does completeness of a category in an adjunction imply completeness for the other? Assume we have an adjunction $(L,R,\varphi):\mathcal{C}\rightarrow\mathcal{D}$ between two categories, and assume also that $\mathcal{D}$ is complete (i.e. closed under limits). Under what assumptions (if any) is $\mathcal{C}$ also complete?
My idea is: say we have a diagram $F:I\rightarrow\mathcal{C}$ (where $I$ denotes some small category). Then $LF$ has a limit $x\in\mathcal{D}$ (by completeness of $\mathcal{D}$). Now we can take $Rx$, and by adjointness it should be very close to a limit for $F$.
 A: In general, it is false, and there are lots of counterexamples. If $C$ has some terminal object $1$, then the unique functor $C \to \{1\}$ has a right adjoint, given by $1 \mapsto 1$. Whereas $\{1\}$ is complete and cocomplete, $C$ is not in general.
However, the following is true (also well-known and quite useful): Let $L : C \to D$ be left adjoint to $R : D \to C$, and $L$ is fully faithful. If $D$ is complete, then also $C$ is complete. Dually: If $R$ is fully faithful and $C$ is cocomplete, then also $D$ is cocomplete. In other words, coreflective subcategories of complete categories are complete, and reflective subcategories of cocomplete categories are cocomplete.
Namely, let $X : I \to C$ be a diagram. Choose a limit $\{L X_i \to L Y\}$ in $D$. Then the natural bijections
$\lim_i \hom(T,X_i) \cong \lim_i \hom(LT,L X_i) \cong \hom(LT,LY) \cong \hom(T,RLY)$
show that $\{X_i \to RLX_i \to RLY\}$ is a limit in $C$.
Example: The category $C$ of torsion abelian groups is complete. To see this, let $D$ be the category of groups, $L$ the inclusion, and $R$ the functor which maps each abelian group $A$ to its torsion subgroup $t(A)$. Then $L$ is left adjoint to $R$ and  fully faithful by definition, and $D$ is complete. Hence, also $C$ is complete. The limit of a diagram of torsion abelian groups is the torsion subgroup of the limit of the underlying abelian groups. For example, $\prod^C_p \mathbb{Z}/p$ becomes $\oplus^D_p \mathbb{Z}/p$.
Dual example: The category $D$ of torsionfree abelian groups is cocomplete. Take $C$ to be the category of groups, $R$ the inclusion and $L$ the functor which maps an abelian group $A$ to its maximal torsionfree quotient, i.e. $A/t(A)$. The colimit of a diagram of torsionfree abelian groups is the maximal torsionfree quotient of the colimit of the underlying abelian groups. For example, $2 : \mathbb{Z} \to \mathbb{Z}$ is an epimorphism in $D$, since its cokernel is the maximal torsionfree quotient of $\mathbb{Z}/2$, which is zero.
Another general and even more basic observation is the following: Let $L,R$ be as above and assume that $L$ is fully faithful. If $D$ is cocomplete, and the colimit of a diagram in $D$ whose objects lie in the image of $C$ also lies in the image of $C$, then $C$ is cocomplete, and actually $L$ creates colimits. This produces, for example, colimits of torsion abelian groups (just take the colimit of the underlying abelian groups). Again there is a dual statement, which gives, for example, limits of torsionfree abelian groups.
