Limits in functor categories

Question

Let $C,C’,D$ be categories and $u:C\to C’$ be a functor. The functor $u^*:\mathbf{Hom}(C’^\circ,D)\to\mathbf{Hom}(C^\circ,D)$ that sends a functor $G$ to $G\circ u$ commutes with limits and colimits, according to SGA 4. Does this presume that $D$ is (co)complete? I know when $D$ is (co)complete, (co)limits are computed pointwise. If not, some (co)limits can exist ‘by accident.’

Answer 1

Yes, it is best to assume that $\mathcal{D}$ is (co)complete for the purposes of this proposition. Actually, there are two ways to make the proposition precise. You know this one:

Proposition. If $\mathcal{D}$ has limits (resp. colimits) of diagrams of shape $\mathcal{J}$ then $[\mathcal{C}'^\textrm{op}, \mathcal{D}]$ and $[\mathcal{C}^\textrm{op}, \mathcal{D}]$ also have limits (resp. colimits) of diagrams of shape $\mathcal{J}$ and $u^* : [\mathcal{C}'^\textrm{op}, \mathcal{D}] \to [\mathcal{C}^\textrm{op}, \mathcal{D}]$ preserves those limits (resp. colimits).

But here is another one:

Proposition. If, for every object $c'$ in $\mathcal{C}'$, $\mathcal{D}$ has colimits (resp. limits) of diagrams of shape $(u \downarrow c')$ (resp. $(c' \downarrow u)$), then $u^* : [\mathcal{C}'^\textrm{op}, \mathcal{D}] \to [\mathcal{C}^\textrm{op}, \mathcal{D}]$ has a left (resp. right) adjoint.

Notice that in this proposition, existence of colimits (resp. limits) of particular shapes implies preservation of limits (resp. colimits) of arbitrary shapes, which may or may not be componentwise!