Suppose $\mathcal{C}$ and $\mathcal{D}$ are categories. A contravariant functor from $\mathcal{C}$ to $\mathcal{D}$ can be defined as a covariant functor $F:\mathcal{C}^{op} \rightarrow \mathcal{D}$, or equivalently $F:\mathcal{C} \rightarrow \mathcal{D}^{op}$. I can see how are they related, and the equivalence is nicely explained in this page "Alternative Definition of Contravariant Functor", but something is still bothering me.
Using the formal definition we end up in the category $\mathcal{D}$, but with the latter we are in the category $\mathcal{D}^{op}$. So for example, if $\mathcal{D}$ is the category of sets, then using the formal definition we are now in the world of sets and functions which is nice, while using the latter definition we are now in the world of complete atomic boolean algebras and complete morphisms which is not so nice. This makes me feel uneasy because using different definition seems to land us in different 'world' (though the 'worlds' are dual). Can someone please help me understand this better?