Definition 5.1.5.1.1 (01NU): The Left Unitor of $\times $

The left unitor of the product of sets is the natural isomorphism

whose component

\[ \lambda ^{\mathsf{Sets}}_{X} \colon \mathrm{pt}\times X \overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }X \]

at $X\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$ is given by

\[ \lambda ^{\mathsf{Sets}}_{X}(\star ,x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x \]

for each $(\star ,x)\in \mathrm{pt}\times X$.

Invertibility

The inverse of $\lambda ^{\mathsf{Sets}}_{X}$ is the morphism

\[ \lambda ^{\mathsf{Sets},-1}_{X}\colon X\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathrm{pt}\times X \]

defined by

\[ \lambda ^{\mathsf{Sets},-1}_{X}(x)\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(\star ,x) \]

for each $x\in X$. Indeed:

•
Invertibility I. We have

\begin{align*} [\lambda ^{\mathsf{Sets},-1}_{X}\circ \lambda ^{\mathsf{Sets}}_{X}](\mathrm{pt},x) & = \lambda ^{\mathsf{Sets},-1}_{X}(\lambda ^{\mathsf{Sets}}_{X}(\mathrm{pt},x))\\ & = \lambda ^{\mathsf{Sets},-1}_{X}(x)\\ & = (\mathrm{pt},x)\\ & = [\operatorname {\mathrm{id}}_{\mathrm{pt}\times X}](\mathrm{pt},x) \end{align*}

for each $(\mathrm{pt},x)\in \mathrm{pt}\times X$, and therefore we have

\[ \lambda ^{\mathsf{Sets},-1}_{X}\circ \lambda ^{\mathsf{Sets}}_{X}=\operatorname {\mathrm{id}}_{\mathrm{pt}\times X}. \]
•
Invertibility II. We have

\begin{align*} [\lambda ^{\mathsf{Sets}}_{X}\circ \lambda ^{\mathsf{Sets},-1}_{X}](x) & = \lambda ^{\mathsf{Sets}}_{X}(\lambda ^{\mathsf{Sets},-1}_{X}(x))\\ & = \lambda ^{\mathsf{Sets},-1}_{X}(\mathrm{pt},x)\\ & = x\\ & = [\operatorname {\mathrm{id}}_{X}](x) \end{align*}

for each $x\in X$, and therefore we have

\[ \lambda ^{\mathsf{Sets}}_{X}\circ \lambda ^{\mathsf{Sets},-1}_{X}=\operatorname {\mathrm{id}}_{X}. \]

Therefore $\lambda ^{\mathsf{Sets}}_{X}$ is indeed an isomorphism.

Naturality

We need to show that, given a function $f\colon X\to Y$, the diagram

commutes. Indeed, this diagram acts on elements as

and hence indeed commutes. Therefore $\lambda ^{\mathsf{Sets}}$ is a natural transformation.

Being a Natural Isomorphism

Since $\lambda ^{\mathsf{Sets}}$ is natural and $\lambda ^{\mathsf{Sets},-1}$ is a componentwise inverse to $\lambda ^{\mathsf{Sets}}$, it follows from Chapter 11: Categories, Item 2 of Proposition 11.9.7.1.2 that $\lambda ^{\mathsf{Sets},-1}$ is also natural. Thus $\lambda ^{\mathsf{Sets}}$ is a natural isomorphism.

The Clowder Project

5.1.5 The Left Unitor