Let $f\colon A\to B$ and $g\colon B\to C$ be morphisms of $\mathcal{C}$.
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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 $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (B,C\webright )$.
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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 $\phi \in \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (A,B\webright )$.
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1.
Interaction Between Precomposition and Postcomposition. We have
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2.
Interaction With Composition I. We have
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3.
Interaction With Composition II. We have
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4.
Interaction With Composition III. We have
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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*}
11.1.4 Precomposition and Postcomposition
Let $\mathcal{C}$ be a category and let $A,B,C\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$.
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 $\mathcal{C}$.
Proof of Proposition 11.1.4.1.2.
Item 1: Interaction Between Precomposition and Postcomposition
Clear.
Item 2: Interaction With Composition I
Clear.
Item 3: Interaction With Composition II
Clear.
Item 4: Interaction With Composition III
Clear.
Item 5: Interaction With Identities
Clear.
