The end of the hom-bifunctor is the commutative monoid of natural endomorphisms of the identity functor

Given: A small category $\mathcal{C}$ and its identity functor.

A category is small when both $\mathrm{Ob}(\mathcal{C})$ and $\mathrm{Mor}(\mathcal{C})$ are sets; a small category is locally small (Small, locally small, and large categories).

For a small source category and a locally small target, the set of natural transformations is an end of the hom-bifunctor of the values: $\mathrm{Nat}(F,G)={\int }_c\mathcal{D}(Fc,Gc)$, the terminal wedge being evaluation (For a small source category, the set of natural transformations is an end of the hom-bifunctor of the values).

The hom-assignment sends $(a,b)$ to $\mathcal{C}(a,b)$, and a morphism of the product category consisting of $h:a^{\mathrm{\prime }}\to a$ and $u:b\to b^{\mathrm{\prime }}$ acts by $\mathcal{C}(h,u):\mathcal{C}(a,b)⟶\mathcal{C}(a^{\mathrm{\prime }},b^{\mathrm{\prime }}),f⟼u\circ f\circ h$ (The covariant and contravariant hom-assignments and the hom-bifunctor of a locally small category).

The functor category $[\mathcal{C},\mathcal{D}]$ has functors $\mathcal{C}\to \mathcal{D}$ as objects and natural transformations as morphisms (Functor category $[\mathcal{C},\mathcal{D}]$).

The identity natural transformation $1_F$ has components $1_{FA}$, and the vertical composite of $\alpha :F\Rightarrow G$ and $\beta :G\Rightarrow H$ is given componentwise by $(\beta \circ \alpha {)}_A={\beta }_A\circ {\alpha }_A$ (Identity natural transformation and vertical composition).

The horizontal composite of $\alpha :F\Rightarrow G:\mathcal{C}\to \mathcal{D}$ and $\beta :H\Rightarrow L:\mathcal{D}\to \mathcal{E}$ has components $(\beta \ast \alpha {)}_A={\beta }_{GA}\circ H({\alpha }_A)=L({\alpha }_A)\circ {\beta }_{FA}$ (Whiskering and horizontal composition of natural transformations).

A monoid is a set with an associative operation and a two-sided identity $e$, so that $e\ast x=x=x\ast e\text{for every }x\in M\mathrm{.}$ It is commutative when the operation is (Semigroup and monoid).

Whenever the expressions are defined, $({\beta }^{\mathrm{\prime }}\circ \beta )\ast ({\alpha }^{\mathrm{\prime }}\circ \alpha )=({\beta }^{\mathrm{\prime }}\ast {\alpha }^{\mathrm{\prime }})\circ (\beta \ast \alpha )$ (Horizontal and vertical composition of natural transformations satisfy the interchange law).

If a set carries two unital binary operations with the same unit satisfying $(a\circ b)\ast (c\circ d)=(a\ast c)\circ (b\ast d)$, then the operations coincide and their common operation is commutative (Eckmann–Hilton: two unital operations satisfying interchange coincide and are commutative).