The case for left and right adjoint functors commuting with colimits and limits respectively when the functors are presheaves.

Question

Presheaves $F:\mathcal{C}^{opp} \rightarrow \mathcal{D}$, $G:\mathcal{D}^{opp} \rightarrow \mathcal{C}$ are called left (resp. right) adjoint, if there exist a natural bijection (in $C \in \text{Ob}\ \mathcal{C}$, $D \in \text{Ob}\ \mathcal{D}$)$$\text{Hom}_{\mathcal{C}}(G(D),C) \simeq \text{Hom}_{\mathcal{D}}(F(C),D) \ \ (\text{resp. }\ \text{Hom}_{\mathcal{C}}(C,G(D)) \simeq \text{Hom}_{\mathcal{D}}(D,F(C)))$$

Now, I know that left adjoint functor commutes with colimits and right adjoint functor commutes with limits. I wanted to know if anything changes if we consider functors of the sort above.

Answer 1

A functor $G:\mathcal{D}^{op}\to\mathcal{C}$ can instead be considered as a functor $G^{op}:\mathcal{D}\to\mathcal{C}^{op}$, and $F$ and $G$ are left adjoint iff $F$ is left adjoint to $G^{op}$ in the usual sense. So this means $F$ preserves colimits and $G^{op}$ preserves limits, and the latter condition is equivalent to $G$ preserving colimits.

To be clear, when we say "$F$ preserves colimits", that means it turns colimits in $\mathcal{C}^{op}$ into colimits in $\mathcal{D}$. Thinking of $F$ instead as a contravariant functor from $\mathcal{C}$ to $\mathcal{D}$, this means it turns limits in $\mathcal{C}$ into colimits in $\mathcal{D}$. Similarly, $G$ will turn limits in $\mathcal{D}$ into colimits in $\mathcal{C}$.

Of course, the dual story holds when $F$ and $G$ are right adjoint: they both preserve limits, meaning that when you consider them as contravariant functors, they turn colimits into limits.