Functors and natural transformations

A covariant functor $F:\mathcal C\to\mathcal D$ chooses an object $F(A)$ for every object and an arrow $F(f):F(A)\to F(B)$ for every $f:A\to B$. It must preserve identity arrows and composites. Giving only an object assignment is insufficient: choosing dimensions of vector spaces, for example, does not by itself specify what should happen to all linear maps. A functor can identify different objects or different arrows, but any identifications must respect the composition laws.

The opposite category reverses arrows and reverses their composition order. A contravariant functor on $\mathcal C$ is an ordinary covariant functor on $\mathcal C^{\mathrm{op}}$. For instance, fixing an object $A$ makes $\mathcal C(A,-)$ covariant by postcomposition, whereas $\mathcal C(-,A)$ is contravariant by precomposition. Reading source and target types before writing a formula prevents these two Hom constructions from being interchanged.

For parallel functors $F,G:\mathcal C\to\mathcal D$, a natural transformation $\alpha:F\Rightarrow G$ consists of components $\alpha_A:F(A)\to G(A)$. For every arrow $f:A\to B$, following the component and then $G(f)$ must agree with following $F(f)$ and then the component at $B$. Choosing components independently usually fails. Even when the component sets have the same cardinality, naturality can force particular values or prohibit a family altogether.

When the source is an arrow category with just two objects and one nonidentity arrow, a single square controls naturality. For a larger category, all arrows are required; it is enough to check generators only after their relations and the functor laws are established. Natural transformations compose componentwise. If every component is an isomorphism, their inverses are also natural, as follows by multiplying the naturality equation by inverses on the appropriate sides.

Consider the functor $Q$ on Set that sends $X$ to $X\times X$ and sends a function $f$ to its coordinatewise application. The swap maps $\tau_X(x_1,x_2)=(x_2,x_1)$ form a natural transformation from $Q$ to itself. Applying a function to both coordinates and swapping produces exactly the same pair as swapping first and applying the function afterward. This argument works for arbitrary sets and functions, including noninjective maps and empty sets.

Testing a few finite functions can reveal a mistake, but it cannot replace the general calculation establishing naturality on Set. By contrast, in a genuinely finite source category, enumerating every component and every arrow is a complete test. The distinction matters in applications: a coordinate-dependent recipe may accidentally commute with a restricted family of maps while failing for a larger family. The category of allowed transformations is part of the claim being made.