Item 2 (00XU) — The Clowder Project

Table of Contents
Part 4: Categories
Chapter 11: Categories
Section 11.1: Categories
Subsection 11.1.4: Precomposition and Postcomposition
Item 2 (cite)

11.1.4 Precomposition and Postcomposition

Let $\mathcal{C}$ be a category and let $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.

Definition 11.1.4.1.1 ▶ Precomposition and Postcomposition Functions

Let $f\colon A\to B$ and $g\colon B\to C$ be morphisms of $\mathcal{C}$.

1.
The precomposition function associated to $f$ is the function

\[ f^{*} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (B,C\webright ) \to \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,C\webright ) \]

defined by

\[ f^{*}\webleft (\phi \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\phi \circ f \]

for each $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (B,C\webright )$.

2.
The postcomposition function associated to $g$ is the function

\[ g_{*} \colon \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright ) \to \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,C\webright ) \]

defined by

\[ g_{*}\webleft (\phi \webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ \phi \]

for each $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright )$.

Proposition 11.1.4.1.2 ▶ Properties of Pre/Postcomposition

Let $A,B,C,D\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$ and let $f\colon A\to B$ and $g\colon B\to C$ be morphisms of $\mathcal{C}$.

1.
Interaction Between Precomposition and Postcomposition. We have
2.
Interaction With Composition I. We have
3.
Interaction With Composition II. We have
4.
Interaction With Composition III. We have
5.
Interaction With Identities. We have

\begin{align*} \webleft (\operatorname {\mathrm{id}}_{A}\webright )^{*} & = \operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright )},\\ \webleft (\operatorname {\mathrm{id}}_{B}\webright )_{*} & = \operatorname {\mathrm{id}}_{\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright )}. \end{align*}