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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 $C$.

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 $C$.

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 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.

For a smooth manifold $M$ over a field $F$, the cotangent space at $p$ is defined as $\operatorname{Hom}(T_pM, F)$. The earlier $\mathbb{R}^n$ example is recovered by taking $M = \mathbb{R}^n$ and $F = \mathbb{R}$.

Conditions: $M$ is a smooth manifold.; $F$ is the field over which the manifold is considered.

For a smooth manifold $M$ over a field $F$, the cotangent space at $p$ is defined as $\operatorname{Hom}(T_pM, F)$. The earlier $\mathbb{R}^n$ example is recovered by taking $M = \mathbb{R}^n$ and $F = \mathbb{R}$.

Conditions: $M$ is a smooth manifold.; $F$ is the field over which the manifold is considered.