Is the opposite functor full and bijective on objects?

Question

Is that true that the opposite functor that is

$$\mathbf{O}:\mathcal{F}\to\mathcal{F}^{op}$$

is bijective on objects and full?

Answer 1

$\newcommand{\F}{\mathcal{F}}\newcommand{\op}{\mathrm{op}}\newcommand{\O}{\mathbf{O}}\newcommand{\Cat}{\mathrm{Cat}}$I think you are misunderstanding the meaning of “the opposite functor”. In general, there is no canonical functor $\F \to \F^\op$, for arbitrary categories $\F$. Instead, you probably want the functor

$$ \O : \Cat \to \Cat $$

defined (on objects) by $\O(\F) = \F^\op$.

With the correct functor, hopefully you will find the question more tractable. Let me know in comments if you need further hints.