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

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

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

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

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

for each $x\in X$. Indeed:

•
Invertibility I. We have

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

for each $\webleft (\mathrm{pt},x\webright )\in \mathrm{pt}\times X$, 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*} \webleft [\lambda ^{\mathsf{Sets}}_{X}\circ \lambda ^{\mathsf{Sets},-1}_{X}\webright ]\webleft (x\webright ) & = \lambda ^{\mathsf{Sets}}_{X}\webleft (\lambda ^{\mathsf{Sets},-1}_{X}\webleft (x\webright )\webright )\\ & = \lambda ^{\mathsf{Sets},-1}_{X}\webleft (\mathrm{pt},x\webright )\\ & = x\\ & = \webleft [\operatorname {\mathrm{id}}_{X}\webright ]\webleft (x\webright ) \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