Right adjoints preserve every limit that exists

Let ${\mu }_j:c\to G{D}_j$ be any cone over $GD$, and define ${\bar{\mu }}_j:={\epsilon }_{D_j}\circ F({\mu }_j):Fc\to D_j$.

For $a:j\to k$ in $\mathcal{J}$, naturality of $\epsilon$ and the cone equation for $\mu$ give $D(a){\bar{\mu }}_j={\epsilon }_{D_k}FGD(a)F({\mu }_j)={\epsilon }_{D_k}F(GD(a){\mu }_j)={\bar{\mu }}_k$, so $\bar{\mu }$ is a cone over $D$.

By [F1], there is a unique $h:Fc\to L$ with ${\lambda }_{j}h={\bar{\mu }}_j$ for every $j$. Define $\widehat{h}:=G(h){\eta }_c:c\to GL$.

For each $j$, $G({\lambda }_j)\widehat{h}=G({\bar{\mu }}_j){\eta }_c=G({\epsilon }_{D_j})GF({\mu }_j){\eta }_c=G({\epsilon }_{D_j}){\eta }_{G{D}_j}{\mu }_j={\mu }_j$, by naturality of $\eta$ and the second triangle identity.

If $k:c\to GL$ also satisfies $G({\lambda }_j)k={\mu }_j$, then ${\epsilon }_{L}F(k):Fc\to L$ mediates $\bar{\mu }$ by naturality of $\epsilon$. Uniqueness in [F1] makes it $h$, and naturality of $\eta$ together with the second triangle identity $G({\epsilon }_L){\eta }_{GL}=1_{GL}$ give $k=G({\epsilon }_L)GF(k){\eta }_c=G({\epsilon }_L){\eta }_{GL}k$, that is $k=G(h){\eta }_c=\widehat{h}$.

Thus every cone over $GD$ factors uniquely through $(GL,G\lambda )$, so it is limiting by [F1]. The construction also covers the empty diagram, where there are no leg equations, and [F2] says exactly that $G$ preserves the limit.