Given categories $\mathcal{C}$ and $\mathcal{D}$, the functor category $\mathcal{D}^\mathcal{C}$ (or $[\mathcal{C}, \mathcal{D}]$) has:

• Objects: functors $F: \mathcal{C} \to \mathcal{D}$
• Morphisms: natural transformations between functors (to be defined precisely later)

This construction shows that functors themselves form a category, revealing the "higher-order" nature of category theory.

Let $\mathcal{C}$ be a category and $A$ an object in $\mathcal{C}$. The slice category $\mathcal{C}/A$ (or over category) has:

• Objects: pairs $(B, f)$ where $B$ is an object of $\mathcal{C}$ and $f: B \to A$ is a morphism
• Morphisms: from $(B, f)$ to $(C, g)$ are morphisms $h: B \to C$ in $\mathcal{C}$ such that $g \circ h = f$

The dual construction gives the coslice category $A/\mathcal{C}$ (or under category).

1. Free group functor $F: \textbf{Set} \to \textbf{Grp}$ assigns to each set $X$ the free group $F(X)$ generated by $X$

2. Abelianization functor $\text{Ab}: \textbf{Grp} \to \textbf{Ab}$ sends each group $G$ to its abelianization $G/[G,G]$

3. Tensor product functor $- \otimes_R M: \textbf{Mod}_R \to \textbf{Mod}_R$ for a fixed $R$-module $M$

4. Hom functor $\text{Hom}_R(M, -): \textbf{Mod}_R \to \textbf{Ab}$ for a fixed $R$-module $M$ (covariant in the second variable)

1. Fundamental group functor $\pi_1: \textbf{Top}_* \to \textbf{Grp}$ from pointed topological spaces to groups

2. Singular homology functors $H_n: \textbf{Top} \to \textbf{Ab}$ for each $n \geq 0$

3. Singular cohomology functors $H^n: \textbf{Top}^{\text{op}} \to \textbf{Ab}$ (contravariant)

4. Path space functor $P: \textbf{Top} \to \textbf{Top}$ sending a space $X$ to its path space $PX = \{f: [0,1] \to X \mid f \text{ continuous}\}$

Fix an object $A$ in category $\mathcal{C}$.

1. The covariant hom-functor $\text{Hom}_\mathcal{C}(A, -): \mathcal{C} \to \textbf{Set}$ sends:

   • Object $B$ to the set $\text{Hom}_\mathcal{C}(A, B)$
   • Morphism $f: B \to C$ to the function $f_*: \text{Hom}_\mathcal{C}(A, B) \to \text{Hom}_\mathcal{C}(A, C)$ given by $f_*(g) = f \circ g$

2. The contravariant hom-functor $\text{Hom}_\mathcal{C}(-, A): \mathcal{C}^{\text{op}} \to \textbf{Set}$ sends:

   • Object $B$ to the set $\text{Hom}_\mathcal{C}(B, A)$
   • Morphism $f: B \to C$ to the function $f^*: \text{Hom}_\mathcal{C}(C, A) \to \text{Hom}_\mathcal{C}(B, A)$ given by $f^*(g) = g \circ f$

For any category $\mathcal{C}$, the diagonal functor $\Delta: \mathcal{C} \to \mathcal{C} \times \mathcal{C}$ sends:

• Object $A$ to $(A, A)$
• Morphism $f: A \to B$ to $(f, f): (A, A) \to (B, B)$

This functor plays a crucial role in defining products and coproducts via limits and colimits.