8.4.1 Foundations
Let $\mathcal{C}$ and $\mathcal{D}$ be categories.
A functor $F\colon \mathcal{C}\to \mathcal{D}$ from $\mathcal{C}$ to $\mathcal{D}$ consists of:
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Action on Objects. A map of sets
\[ F \colon \text{Obj}\webleft (\mathcal{C}\webright ) \to \text{Obj}\webleft (\mathcal{D}\webright ), \]
called the action on objects of $F$.
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Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, a map
\[ F_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{D}}\webleft (F\webleft (A\webright ),F\webleft (B\webright )\webright ), \]
called the action on morphisms of $F$ at $\webleft (A,B\webright )$.
satisfying the following conditions:
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Preservation of Identities. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram
commutes, i.e. we have
\[ F\webleft (\text{id}_{A}\webright ) = \text{id}_{F\webleft (A\webright )}. \]
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Preservation of Composition. For each $A,B,C\in \text{Obj}\webleft (\mathcal{C}\webright )$, the diagram
commutes, i.e. for each composable pair $\webleft (g,f\webright )$ of morphisms of $\mathcal{C}$, we have
\[ F\webleft (g\circ f\webright ) = F\webleft (g\webright )\circ F\webleft (f\webright ). \]
Let $\mathcal{C}$ and $\mathcal{D}$ be categories, and write $\mathcal{C}^{\mathsf{op}}$ for the opposite category of $\mathcal{C}$ of 
.
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Given a functor
\[ F\colon \mathcal{C}\to \mathcal{D}, \]
we also write $F_{A}$ for $F\webleft (A\webright )$.
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Given a functor
\[ F\colon \mathcal{C}^{\mathsf{op}}\to \mathcal{D}, \]
we also write $F^{A}$ for $F\webleft (A\webright )$.
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Given a functor
\[ F\colon \mathcal{C}\times \mathcal{C}\to \mathcal{D}, \]
we also write $F_{A,B}$ for $F\webleft (A,B\webright )$.
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Given a functor
\[ F\colon \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\to \mathcal{D}, \]
we also write $F^{A}_{B}$ for $F\webleft (A,B\webright )$.
We employ a similar notation for morphisms, writing e.g. $F_{f}$ for $F\webleft (f\webright )$ given a functor $F\colon \mathcal{C}\to \mathcal{D}$.
Following the notation $[\mspace {-3mu}[x\mapsto f\webleft (x\webright )]\mspace {-3mu}]$ for a function $f\colon X\to Y$ introduced in Chapter 1: Sets, Notation 1.1.1.1.2, we will sometimes denote a functor $F\colon \mathcal{C}\to \mathcal{D}$ by
\[ F\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[A\mapsto F\webleft (A\webright )]\mspace {-3mu}], \]
specially when the action on morphisms of $F$ is clear from its action on objects.
The identity functor of a category $\mathcal{C}$ is the functor $\text{id}_{\mathcal{C}}\colon \mathcal{C}\to \mathcal{C}$ where
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Action on Objects. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, we have
\[ \text{id}_{\mathcal{C}}\webleft (A\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A. \]
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Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the action on morphisms
\[ \webleft (\text{id}_{\mathcal{C}}\webright )_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \underbrace{\textup{Hom}_{\mathcal{C}}\webleft (\text{id}_{\mathcal{C}}\webleft (A\webright ),\text{id}_{\mathcal{C}}\webleft (B\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )} \]
of $\text{id}_{\mathcal{C}}$ at $\webleft (A,B\webright )$ is defined by
\[ \webleft (\text{id}_{\mathcal{C}}\webright )_{A,B} \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )}. \]
Preservation of Identities
We have $\text{id}_{\mathcal{C}}\webleft (\text{id}_{A}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{A}$ for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$ by definition.
Preservation of Compositions
For each composable pair $A\xrightarrow {f}B\xrightarrow {g}B$ of morphisms of $\mathcal{C}$, we have
\begin{align*} \text{id}_{\mathcal{C}}\webleft (g\circ f\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ f\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{id}_{\mathcal{C}}\webleft (g\webright )\circ \text{id}_{\mathcal{C}}\webleft (f\webright ). \end{align*}
This finishes the proof.
The composition of two functors $F\colon \mathcal{C}\to \mathcal{D}$ and $G\colon \mathcal{D}\to \mathcal{E}$ is the functor $G\circ F$ where
- Action on Objects. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, we have
\[ \webleft [G\circ F\webright ]\webleft (A\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}G\webleft (F\webleft (A\webright )\webright ). \]
- Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$, the action on morphisms
\[ \webleft (G\circ F\webright )_{A,B} \colon \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright ) \to \textup{Hom}_{\mathcal{E}}\webleft (G_{F_{A}},G_{F_{B}}\webright ) \]
of $G\circ F$ at $\webleft (A,B\webright )$ is defined by
\[ \webleft [G\circ F\webright ]\webleft (f\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}G\webleft (F\webleft (f\webright )\webright ). \]
Preservation of Identities
For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, we have
\begin{align*} G_{F_{\text{id}_{A}}} & = G_{\text{id}_{F_{A}}}\tag {functoriality of $F$}\\ & = \text{id}_{G_{F_{A}}}.\tag {functoriality of $G$} \end{align*}
Preservation of Composition
For each composable pair $\webleft (g,f\webright )$ of morphisms of $\mathcal{C}$, we have
\begin{align*} G_{F_{g\circ f}} & = G_{F_{g}\circ F_{f}}\tag {functoriality of $F$}\\ & = G_{F_{g}}\circ G_{F_{f}}.\tag {functoriality of $G$} \end{align*}
This finishes the proof.
Let $F\colon \mathcal{C}\to \mathcal{D}$ be a functor.
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Preservation of Isomorphisms. If $f$ is an isomorphism in $\mathcal{C}$, then $F\webleft (f\webright )$ is an isomorphism in $\mathcal{D}$.
Item 1: Preservation of Isomorphisms
Indeed, we have
\begin{align*} F\webleft (f\webright )^{-1}\circ F\webleft (f\webright ) & = F\webleft (f^{-1}\circ f\webright )\\ & = F\webleft (\text{id}_{A}\webright )\\ & = \text{id}_{F\webleft (A\webright )} \end{align*}
and
\begin{align*} F\webleft (f\webright )\circ F\webleft (f\webright )^{-1} & = F\webleft (f\circ f^{-1}\webright )\\ & = F\webleft (\text{id}_{B}\webright )\\ & = \text{id}_{F\webleft (B\webright )}, \end{align*}
showing $F\webleft (f\webright )$ to be an isomorphism.