A morphism $f: A \to B$ in a category $\mathcal{C}$ is an isomorphism if there exists a morphism $g: B \to A$ such that: $g \circ f = \text{id}_A \quad \text{and} \quad f \circ g = \text{id}_B$

Objects $A$ and $B$ are isomorphic, written $A \cong B$, if there exists an isomorphism between them.

Let $f: A \to B$ be a morphism in category $\mathcal{C}$.

• $f$ is a monomorphism (monic) if for all morphisms $g, h: C \to A$, we have $f \circ g = f \circ h \implies g = h$
• $f$ is an epimorphism (epic) if for all morphisms $g, h: B \to C$, we have $g \circ f = h \circ f \implies g = h$
• $f$ is a section (split monic) if there exists $g: B \to A$ with $g \circ f = \text{id}_A$
• $f$ is a retraction (split epic) if there exists $g: B \to A$ with $f \circ g = \text{id}_B$

A subcategory $\mathcal{D}$ of a category $\mathcal{C}$ consists of:

• A subcollection of objects of $\mathcal{C}$
• For each pair of objects $A, B$ in $\mathcal{D}$, a subcollection of $\text{Hom}_\mathcal{C}(A, B)$
• The same identity morphisms and composition as in $\mathcal{C}$

The subcategory is full if for all objects $A, B$ in $\mathcal{D}$, we have $\text{Hom}_\mathcal{D}(A, B) = \text{Hom}_\mathcal{C}(A, B)$.

Let $F: \mathcal{C} \to \mathcal{D}$ be a functor.

• $F$ is faithful if for all objects $A, B$ in $\mathcal{C}$, the function $F: \text{Hom}_\mathcal{C}(A, B) \to \text{Hom}_\mathcal{D}(F(A), F(B))$ is injective
• $F$ is full if this function is surjective
• $F$ is fully faithful if it is both full and faithful (i.e., the function is bijective)
• $F$ is essentially surjective if for every object $D$ in $\mathcal{D}$, there exists an object $C$ in $\mathcal{C}$ such that $F(C) \cong D$

Given a category $\mathcal{C}$, the opposite category $\mathcal{C}^{\text{op}}$ has:

• The same objects as $\mathcal{C}$
• For objects $A, B$, we have $\text{Hom}_{\mathcal{C}^{\text{op}}}(A, B) = \text{Hom}_\mathcal{C}(B, A)$
• Composition defined by $g \circ_{\text{op}} f = f \circ g$

A contravariant functor $F: \mathcal{C} \to \mathcal{D}$ is equivalent to a covariant functor $F: \mathcal{C}^{\text{op}} \to \mathcal{D}$.

Given categories $\mathcal{C}$ and $\mathcal{D}$, their product category $\mathcal{C} \times \mathcal{D}$ has:

• Objects: pairs $(A, B)$ where $A \in \text{Ob}(\mathcal{C})$ and $B \in \text{Ob}(\mathcal{D})$
• Morphisms: pairs $(f, g): (A, B) \to (A', B')$ where $f: A \to A'$ in $\mathcal{C}$ and $g: B \to B'$ in $\mathcal{D}$
• Composition: $(f', g') \circ (f, g) = (f' \circ f, g' \circ g)$