# Right adjoint functor fully faithful

Let $\mathcal{C}$ be a category of left modules over some ring. Let $F:\mathcal C \rightarrow \mathcal C'$ be a functor with fully faithful right adjoins functor G. Then $\mathcal C'$ has projective limits.

It's easy to prove that $\mathcal C'$ has inductive limits: the fully faithfulness implies that $F$ is a dense functor. Then since left adjoint functor preserves inductive limits, $\mathcal C'$ has inductive limits. But what for projective limits?

There is a more general result about the completeness. Consider a complete category $\mathcal B$ and a full subcategory $\mathcal A$ such that the inclusion functor $i: \mathcal A\rightarrow \mathcal B$ admits a left adjoint $r:\mathcal B\rightarrow \mathcal A$. Then the category $\mathcal A$ is complete.
Sketch of proof: Take a small category $\mathcal D$ and a functor $H: \mathcal D\rightarrow \mathcal A$. Then the composition $i\cdot H$ has a limit $L$ in $\mathcal B$. Denote by $p_D:L\rightarrow i\cdot H D$ the limit morphism for each $D$ in $\mathcal D$. Then we have $\varphi(p_D):rL\rightarrow HD$ where $\varphi$ is the isomorphism in the $\mathrm {Hom}$-set of adjoint pair $(r,i)$. This gives a cone in $\mathcal A$ by the naturality of $\varphi$. Denote by $\eta$ and $\epsilon$ the unit and counit respectively. Since $i$ is fully faithful and $\varphi(p_D)=\epsilon_{HD}\cdot r(p_D)$, we duduce that $rL$ and $\varphi(p_D)$ are the limit of the functor $H$.