Limits and colimits preservation under adjoint equivalence of categories

If $$F\colon\mathcal{A}\to\mathcal{B}$$ is left adjoint to $$U\colon\mathcal{B}\to\mathcal{A}$$, then $$U$$ preserves limits and $$F$$ preserves colimits.

Can we say something more if $$F$$ is left adjoint to $$U$$ and they form an adjoint equivalence between $$\mathcal{A}$$ and $$\mathcal{B}$$?

My question arises from the following statement on Wikipedia:

The functor $$H\colon\mathcal{I}\to\mathcal{C}$$ has limit (or colimit) $$\mathcal{I}$$ if and only if the functor $$FH\colon\mathcal{I}\to\mathcal{D}$$ has limit (or colimit) $$F\mathcal{I}$$.

where $$F\colon\mathcal{C}\to\mathcal{D}$$ is an equivalence. Is $$F$$ preserving both limits and colimits? (in the Wikipedia case, $$F$$ is not part of an adjoint equivalence, but, according to this question, every equivalence can be upgraded to an adjoint equivalence).

An adjoint equivalence between categories $$\mathcal{A}$$ and $$\mathcal{B}$$ is not a pair $$(F,U)$$ of functors between them, but a quadruplet $$(F,U,\eta,\varepsilon)$$, where $$\eta$$ and $$\varepsilon$$ are corresponding natural transformations, such that $$(F,U,\eta,\varepsilon)$$ is an adjunction. Note, that every pair $$(F,U)$$, where $$F$$ and $$U$$ are mutually inverse equivalences of categories, may be completed to such a quadruplet. Therefore, every equivalence of categories is both a left-adjoint and a right-adjoint functor.
When $$F$$ and $$U$$ form an adjoint equivalence, the unit $$\eta:I_{\mathcal{A}}\to UF$$ and counit $$\epsilon : FU\to I_{\mathcal{B}}$$ are isomorphisms; then their inverses $$\epsilon^{-1}:I_{\mathcal{B}}\to FU$$ and $$\eta^{-1}:UF\to I_{\mathcal{A}}$$ are natural transformation, and they satisfy the identities $$F(\eta^{-1})\circ\epsilon^{-1}_F=(\epsilon_F\circ F(\eta))^{-1}=1_F$$ and $$\eta^{-1}_U\circ U(\epsilon^{-1})=(U(\epsilon)\circ \eta_U)^{-1}=1_U.$$ Thus $$U$$ is also left adjoint to $$F$$, so that $$U$$ preserve colimits and $$F$$ preserve limits.