I'm trying to self-learn category theory using Emily Riehl's Category Theory in Context. Exercise 1.3.v asks the difference between a functor $\mathbf{C}\to \mathbf{D}$ and a functor $\mathbf{C}^{op} \to \mathbf{D}^{op}$, and I find that question incredibly and frustratingly confusing.
I believe there is no difference other than the way such functors are defined (the second one begins and ends in the opposite categories of $\mathbf{C}$ and $\mathbf{D}$). Say that $\mathbf{F} : \mathbf{C}^{op} \to \mathbf{D}^{op}$ is a functor. For a morphism $f: A \to B$ in $\mathbf{C}$ we have that $f: B \to A$ is a morphism in $\mathbf{C}^{op}$, so that $\mathbf{F}f : \mathbf{F} B \to \mathbf{F} A$ is a morphism in $\mathbf{D}^{op}$, which itself translates to a morphism $\mathbf{F}f : \mathbf{F} A \to \mathbf{F} B$ in $\mathbf{D}$.
However I am having trouble dealing with the image of compositions. Is it okay to say that, by denoting $\circ_{\mathbf{C}}$ and $\circ_{\mathbf{C}^{op}}$ the respective compositions, we have $$\mathbf{F} (g \circ_{\mathbf{C}} f) = \mathbf{F} (f \circ_{\mathbf{C}^{op}} g) = \left( \mathbf{F} f \right)\circ_{\mathbf{D}^{op}} \left(\mathbf{F} g\right) = \left(\mathbf{F} g \right)\circ_{\mathbf{D}} \left(\mathbf{F} f \right)?$$
So essentially, such a functor can be thought as a (covariant) functor between the "non-opposite" categories. Is this an acceptable answer?
Best Answer
You are right. There is a high level view on this matter.
Suppose that $\textbf{CAT}$ is a category of (small with respect to given universe) categories. As you may know $\textbf{CAT}$ is a paradigmatic example of a $2$-category. Morphisms of $2$-categories are called $2$-functors. Then the construction of the opposite category may be viewed as the $2$-functor
$$(-)^{op}:\textbf{CAT} \rightarrow \textbf{CAT}$$
which is an (involutive i.e. $(-)^{op}\cdot (-)^{op}= 1_{\textbf{CAT}}$) isomorphism of $2$-categories. This fact in particular implies that $(-)^{op}$ induces an isomorphism of categories of functors
$$\mathrm{Func}\left(\mathcal{C},\mathcal{D}\right)^{\mathrm{op}} \cong \mathrm{Func}\left(\mathcal{C}^{op},\mathcal{D}^{op}\right)$$
This is a rigorous formulation of the phenomenon you've discovered.