Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense

Since an equivalence is fully faithful, [L1] proves reflection of limits. The quasi-inverse $G$ is also fully faithful and therefore reflects limits.

If $\lambda$ is a limit cone in $\mathcal{C}$, apply $F$. If a cone over $FD$ is given, transport it along the unit and counit of [F1], apply $G$, and factor uniquely through $\lambda$. Transporting the factor back gives existence through $F\lambda$; reflection by $G$ gives uniqueness. Hence $F$ preserves limits.

Let $\mu$ be a limiting cone over $FD$. Applying $G$ and using step 2.1 for $G$ gives a limiting cone over $GFD$. Transport it along the unit $D\cong GFD$ to a limiting cone $\bar{\mu }$ over $D$. Its image is isomorphic as a cone to $\mu$ by the counit and its naturality. Together with reflection from step 1.1, this is creation in [F2].

Applying [L2] to steps 1.1, 2.1, and 3.1 proves preservation, reflection, and isomorphism-invariant creation of colimits. The construction uses the unit and counit isomorphisms, so it supplies no on-the-nose strict lift.