Let $\mathcal{A}$ be an abelian category and $\mathcal{D}$ a triangulated category. Suppose $F: K(\mathcal{A}) \to \mathcal{D}$ is a triangulated functor sending quasi-isomorphisms to isomorphisms. Then $F$ factors uniquely through the derived category: $K(\mathcal{A}) \to D(\mathcal{A}) \to \mathcal{D}$

This universal property characterizes $D(\mathcal{A})$ up to unique equivalence.

For an additive functor $F: \mathcal{A} \to \mathcal{B}$ between abelian categories:

1. If $\mathcal{A}$ has enough injectives, the right derived functor $RF: D^+(\mathcal{A}) \to D^+(\mathcal{B})$ exists
2. If $\mathcal{A}$ has enough projectives, the left derived functor $LF: D^-(\mathcal{A}) \to D^-(\mathcal{B})$ exists

These are characterized by universal properties among triangulated functors.

For a ring $R$, the derived tensor product and derived Hom are: $-\otimes_R^L -: D^-(\text{Mod}_R) \times D^-(\text{Mod}_R) \to D^-(\text{Mod}_R)$ $R\underline{\text{Hom}}_R(-, -): D^+(\text{Mod}_R)^{\text{op}} \times D^+(\text{Mod}_R) \to D^+(\text{Mod}_R)$

These satisfy the adjunction: $\text{Hom}_{D(R)}(X \otimes_R^L Y, Z) \cong \text{Hom}_{D(R)}(X, R\underline{\text{Hom}}_R(Y, Z))$

For a proper morphism $f: X \to Y$ of schemes, there exists a right adjoint $f^!: D^+_{\text{qcoh}}(Y) \to D^+_{\text{qcoh}}(X)$ to $Rf_*$: $\text{Hom}_{D(Y)}(Rf_*\mathcal{F}, \mathcal{G}) \cong \text{Hom}_{D(X)}(\mathcal{F}, f^!\mathcal{G})$

This generalizes Serre duality and is fundamental in algebraic geometry.

For a morphism $f: X \to Y$ of schemes, there are six functors: $f_*, f^*, f_!, f^!, \otimes^L, R\underline{\text{Hom}}$

satisfying various adjunctions and compatibilities. This framework unifies many duality and base change results in algebraic geometry.