Dinatural transformation between functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$

Let $\mathcal{C}$ and $\mathcal{D}$ be categories (Category, object, morphism, domain, codomain, identity, composition, and hom-collection) and let

$P,Q:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}⟶\mathcal{D}$

be functors on the product of ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ with $\mathcal{C}$ (Opposite category ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$, Product category and its projection functors, Covariant functor, identity functor, composite functor, and contravariant functor). A morphism $(a,b)\to (a^{\mathrm{\prime }},b^{\mathrm{\prime }})$ of ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ is a pair $(g,h)$ in which $h:b\to b^{\mathrm{\prime }}$ is a morphism of $\mathcal{C}$ and $g$ is a morphism of ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}$ from $a$ to $a^{\mathrm{\prime }}$, that is, a morphism $a^{\mathrm{\prime }}\to a$ of $\mathcal{C}$. Thus a morphism $f:c\to c^{\mathrm{\prime }}$ of $\mathcal{C}$ supplies the four morphisms

$(f,1_c):(c^{\mathrm{\prime }},c)\to (c,c),(1_{c^{\mathrm{\prime }}},f):(c^{\mathrm{\prime }},c)\to (c^{\mathrm{\prime }},c^{\mathrm{\prime }}),(1_c,f):(c,c)\to (c,c^{\mathrm{\prime }}),(f,1_{c^{\mathrm{\prime }}}):(c^{\mathrm{\prime }},c^{\mathrm{\prime }})\to (c,c^{\mathrm{\prime }})\mathrm{.}$

A dinatural transformation $\alpha :P\to Q$ is a family of morphisms

${\alpha }_c:P(c,c)⟶Q(c,c)(c\in \mathrm{Ob}(\mathcal{C}))$

of $\mathcal{D}$ such that every morphism $f:c\to c^{\mathrm{\prime }}$ of $\mathcal{C}$ satisfies the dinaturality equation

$Q(1_c,f)\circ {\alpha }_c\circ P(f,1_c)=Q(f,1_{c^{\mathrm{\prime }}})\circ {\alpha }_{c^{\mathrm{\prime }}}\circ P(1_{c^{\mathrm{\prime }}},f)$

between morphisms $P(c^{\mathrm{\prime }},c)\to Q(c,c^{\mathrm{\prime }})$. The morphism ${\alpha }_c$ is the component of $\alpha$ at $c$.

The two sides pass through the six objects $P(c^{\mathrm{\prime }},c)$, $P(c,c)$, $Q(c,c)$, $P(c^{\mathrm{\prime }},c^{\mathrm{\prime }})$, $Q(c^{\mathrm{\prime }},c^{\mathrm{\prime }})$ and $Q(c,c^{\mathrm{\prime }})$, so the diagram expressing the dinaturality equation is a six-sided cycle and is called the hexagon:

A dinatural transformation is a family indexed by the objects of $\mathcal{C}$ and constrained by the morphisms of $\mathcal{C}$, exactly as a natural transformation is (Natural transformation and its components); what differs is which constraint. A natural transformation between two functors on ${\mathcal{C}}^{\mathrm{o}\mathrm{p}}\times \mathcal{C}$ is a family indexed by the pairs $(a,b)$, and its naturality equation is imposed for every morphism of the product category. A dinatural transformation has components only on the diagonal, and the dinaturality equation is imposed only for the morphisms of $\mathcal{C}$ that connect two diagonal entries through the two off-diagonal entries displayed above.

At $f=1_c$ all four displayed morphisms of the product category are identities, so the hexagon reads ${\alpha }_c={\alpha }_c$. The identity morphisms of $\mathcal{C}$ therefore impose no condition, and a dinatural transformation carries no analogue of the identity clause of a functor.