We have

\text{Hom}_{\mathcal{C}}(X,\ R\ \varprojlim D) \cong \text{Hom}_{\mathcal{C}}(X, \varprojlim RD)

and

\text{Hom}_{\mathcal{D}}(L \varinjlim X, Y) \cong \text{Hom}_{\mathcal{D}}(\varinjlim LX, Y)

Proof [local-0]

Only prove right adjoints preserve limits here, left adjoints case is dual.

\begin{aligned} \text{Hom}_{\mathcal{C}}(X,\ R\ \varprojlim D) &\cong\;\text{Hom}_\mathcal{D}(L\ X, \varprojlim D) && \text{by adjunction} \\ &\;\cong\;\varprojlim_j\ \text{Hom}_\mathcal{D}(L\ X, D_j) && \text{representable preserves limits} \\ &\;\cong\;\varprojlim_j\ \text{Hom}_\mathcal{C}(X, R\ D_j) && \text{by adjunction} \\ &\;\cong\;\text{Hom}_\mathcal{C}(X, \varprojlim RD) && \text{representable preserves limits} \end{aligned}

Theorem. Left/right adjoints preserve (co)/limits [0IGM]