Let $Cat^{rex}$ be the category whose objects are (small) categories admitting finite colimits and let $Cat^{lex}$ be the category whose objects are (small) categories admitting finite limits. My first question is: $Cat^{lex}$ and $Cat^{rex}$ are equivalent? I suspect that the answer is yes and the the equivalence functor is simply $(-)^{op}:Cat^{lex}\to Cat^{rex}:A\mapsto A^{op}$.
It is possible defined $\infty$-categorical version of the above categories, we denote them by $Cat^{lex}_{\infty}$ and $Cat^{rex}_{\infty}$. My second question is: $Cat^{lex}_{\infty}$ and $Cat^{rex}_{\infty}$ are equivalent? Another time, I suspect that the answer is yes and the equivalence $\infty$-functor is $(-)^{op}:Cat^{lex}_{\infty}\to Cat^{rex}_{\infty}:A\mapsto A^{op}$.
Every comment is welcome.
Yes, and the proof in both cases is not very technical: the functors $(-)^\mathrm{op}\colon\mathrm{Cat}^\mathrm{lex}_\infty\to\mathrm{Cat}^\mathrm{rex}_\infty$ and $(-)^\mathrm{op}\colon\mathrm{Cat}^\mathrm{rex}_\infty\to\mathrm{Cat}^\mathrm{lex}_\infty$ are well-defined (this is the statement that a category has finite limits (and a functor is left exact) iff the opposite category has finite colimits (and the opposite functor is right exact)), and since $(-)^\mathrm{op}\colon\mathrm{Cat}_\infty\to\mathrm{Cat}_\infty$ is an equivalence of $\infty$-categories with itself as inverse, both composites of the two functors $(-)^\mathrm{op}\colon\mathrm{Cat}^\mathrm{lex}_\infty\to\mathrm{Cat}^\mathrm{rex}_\infty$ and $(-)^\mathrm{op}\colon\mathrm{Cat}^\mathrm{rex}_\infty\to\mathrm{Cat}^\mathrm{lex}_\infty$ are homotopic to the respective identity functors (this needs the observation that equivalences of $\infty$-categories are both left and right exact in order to prove that the natural transformations between $(-)^\mathrm{op}\circ(-)^\mathrm{op}$ and $\mathrm{id}$ truly lie in the correct (non-full) subcategories of $\mathrm{Cat}_\infty$). Therefore the latter two functors are equivalences of $\infty$-categories, and this proves the $\infty$-categorical statement. To obtain the $1$-categorical statement, we note that $\mathrm{Cat}_1^\mathrm{lex}$ is a full subcategory of $\mathrm{Cat}_\infty^\mathrm{lex}$ on the $1$-categories. As such, the equivalence $(-)^\mathrm{op}\colon\mathrm{Cat}_\infty^\mathrm{lex}\to\mathrm{Cat}_\infty^\mathrm{rex}$ restricts to a functor $(-)^\mathrm{op}\colon\mathrm{Cat}_1^\mathrm{lex}\to\mathrm{Cat}_\infty^\mathrm{rex}$ that is an equivalence onto its essential image. We know that a category is a $1$-category iff its opposite is, and hence the essential image of the latter functor is $\mathrm{Cat}_1^\mathrm{rex}$, and as such we also get an equivalence $(-)^\mathrm{op}\colon\mathrm{Cat}_1^\mathrm{lex}\to\mathrm{Cat}_1^\mathrm{rex}$ of $(2,1)$-categories (as both $\infty$-categories are actually $(2,1)$-categories).