The Clowder Project

Let $\alpha \colon F\Longrightarrow G$ and $\beta \colon G\Rightarrow H$ be natural transformations.

1. 1.

   Characterisations. The following conditions are equivalent:

   1. (a)

      The natural transformation $\alpha $ is a natural isomorphism.

   2. (b)

      For each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, the morphism $\alpha _{A}\colon F_{A}\to G_{A}$ is an isomorphism.

2. 2.

   Componentwise Inverses of Natural Transformations Assemble Into Natural Transformations. Let $\alpha ^{-1}\colon G\Longrightarrow F$ be a transformation such that, for each $A\in \operatorname {\mathrm{Obj}}(\mathcal{C})$, we have

   \begin{align*} \alpha ^{-1}_{A}\circ \alpha _{A} & = \operatorname {\mathrm{id}}_{F(A)},\\ \alpha _{A}\circ \alpha ^{-1}_{A} & = \operatorname {\mathrm{id}}_{G(A)}. \end{align*}

   Then $\alpha ^{-1}$ is a natural transformation.

3. 3.

   Interaction With Vertical Composition. If $\alpha $ and $\beta $ are natural isomorphisms, then so is $\beta \circ \alpha $.

4. 4.

   Interaction With Horizontal Composition. Let $\alpha \colon F_{1}\Rightarrow G_{1}$ and $\beta \colon F_{2}\Rightarrow G_{2}$ be natural transformations.

   1. (a)

      If $\alpha $ and $\beta $ are natural isomorphisms, then so is $\beta \mathbin {\star }\alpha $.

   2. (b)

      If $\alpha $ is an isomorphism, then so is $\operatorname {\mathrm{id}}_{G_{1}}\mathbin {\star }\alpha \colon G_{1}\circ F_{1}\Rightarrow G_{1}\circ F_{2}$.

   3. (c)

      If $\beta $ is an isomorphism, then so is $\beta \mathbin {\star }\operatorname {\mathrm{id}}_{F_{1}}\colon G_{1}\circ F_{1}\Rightarrow G_{2}\circ F_{1}$.