How is a natural transformation between two functors defined?

A diagram is said to commute if any two directed paths between the same pair of objects yield equal composite morphisms. In the context of natural transformations, this means that the composition of morphisms along one route in the square (e.g., $F(h)$ followed by $\eta_y$) is exactly equal to the composition along the alternative route (e.g., $\eta_x$ followed by $G(h)$).

Conditions: Applies to diagrams formed by morphisms in a category.; The paths must start at the same object and end at the same object.

The category of functors from $C$ to $D$, denoted $\mathcal{D}^\mathcal{C}$, is a category where the objects are the functors from $C$ to $D$, and the morphisms are the natural transformations between these functors.

Conditions: $C$ and $D$ are categories.

The diagram must commute to ensure that the natural transformation is structurally consistent with the functors and the categories they connect. Specifically, it guarantees that transforming an object via $\eta$ and then applying the functor $G$ to a morphism yields the same result as applying the functor $F$ to the morphism first and then transforming via $\eta$.

Conditions: $h$ is a morphism in $C$ from object $x$ to object $y$.; $F$ and $G$ are functors from $C$ to $D$.; $\eta_x$ and $\eta_y$ are components of the natural transformation $\eta$.

The composition of two natural transformations $\eta : F \to G$ and $\gamma : G \to H$ is defined componentwise. For each object $x$ in the source category $C$, the $x$-component of the composite transformation $\gamma \circ \eta$ is the composite morphism $\gamma_x \circ \eta_x$ in the target category $D$.

Conditions: $\eta$ is a natural transformation from $F$ to $G$.; $\gamma$ is a natural transformation from $G$ to $H$.; $x$ is an object in $C$.

A natural transformation $\eta$ between two functors $F$ and $G$ (both mapping from category $C$ to category $D$) is a rule that assigns to every object $x$ in $C$ a morphism $\eta_x$ in $D$, specifically from $F(x)$ to $G(x)$. This morphism $\eta_x$ is called the component of the natural transformation at object $x$.

Conditions: $F$ and $G$ are functors from category $C$ to category $D$.; For each object $x$ in $C$, $\eta_x$ is a morphism in $D$ from $F(x)$ to $G(x)$.; For every morphism $h: x \to y$ in $C$, the commutative square condition $\eta_y \circ F(h) = G(h) \circ \eta_x$ must hold.