The category of right-module endofunctors

Let $(\mathcal{C},\otimes ,\mathbf{1},\alpha ,\lambda ,\rho )$ be a monoidal category (Monoidal category).

The category of right-module endofunctors ${\mathcal{C}}^{\mathrm{\prime }}$ has:

• objects given by pairs $(F,c)$ where $F:\mathcal{C}\to \mathcal{C}$ is a functor (Covariant functor, identity functor, composite functor, and contravariant functor) and $c_{X,Y}:F(X)\otimes Y⟶F(X\otimes Y)$ is a natural isomorphism (Natural isomorphism) in $X$ and $Y$ such that $F({\alpha }_{X,Y,Z})\circ c_{X\otimes Y,Z}\circ (c_{X,Y}\otimes 1_Z)=c_{X,Y\otimes Z}\circ {\alpha }_{F(X),Y,Z}$ and $F({\rho }_X)\circ c_{X,\mathbf{1}}={\rho }_{F(X)};$
• morphisms $(F,c)\to (G,d)$ given by natural transformations $\theta :F\Rightarrow G$ (Natural transformation and its components) satisfying ${\theta }_{X\otimes Y}\circ c_{X,Y}=d_{X,Y}\circ ({\theta }_X\otimes 1_Y)$ for all objects $X,Y$.

Identities and composition are inherited from natural transformations: the identity transformation satisfies the compatibility equation immediately, and if $\theta :(F,c)\Rightarrow (G,d)$ and $\eta :(G,d)\Rightarrow (H,e)$ are compatible, then substituting their two equations shows that $\eta \circ \theta$ is compatible as well. Associativity and the identity laws are therefore inherited from vertical composition of natural transformations.

Thus an object of ${\mathcal{C}}^{\mathrm{\prime }}$ is an endofunctor equipped with a coherent way to slide a tensor factor on the right through the functor.