The free group functor $F : \mathbf{Set} \to \mathbf{Grp}$ is left adjoint to the forgetful functor $U : \mathbf{Grp} \to \mathbf{Set}$:

$\mathrm{Hom}_{\mathbf{Grp}}(F(S), G) \cong \mathrm{Hom}_{\mathbf{Set}}(S, U(G))$

A group homomorphism from the free group on $S$ is determined by a function from $S$ to $U(G)$. The unit $\eta_S : S \to U(F(S))$ is the inclusion of generators.

$F : \mathbf{Set} \to R\text{-}\mathbf{Mod}$ (free module) is left adjoint to the forgetful functor $U$: $\mathrm{Hom}_R(R^{(S)}, M) \cong \mathrm{Hom}_{\mathbf{Set}}(S, U(M))$.

For a ring $R$ and a right $R$-module $M$, the functors $M \otimes_R - : R\text{-}\mathbf{Mod} \to \mathbf{Ab}$ and $\mathrm{Hom}_{\mathbb{Z}}(M, -) : \mathbf{Ab} \to R\text{-}\mathbf{Mod}$ form an adjunction:

$\mathrm{Hom}_{\mathbb{Z}}(M \otimes_R N, A) \cong \mathrm{Hom}_R(N, \mathrm{Hom}_{\mathbb{Z}}(M, A))$

This is the tensor-Hom adjunction, fundamental in homological algebra.

The abelianization functor $\mathrm{ab} : \mathbf{Grp} \to \mathbf{Ab}$ sending $G \mapsto G/[G,G]$ is left adjoint to the inclusion $\iota : \mathbf{Ab} \hookrightarrow \mathbf{Grp}$:

$\mathrm{Hom}_{\mathbf{Ab}}(G^{\mathrm{ab}}, A) \cong \mathrm{Hom}_{\mathbf{Grp}}(G, \iota(A))$

Sheafification $a : \mathbf{PSh}(X) \to \mathbf{Sh}(X)$ is left adjoint to the inclusion $\iota : \mathbf{Sh}(X) \hookrightarrow \mathbf{PSh}(X)$. The unit $\eta_{\mathcal{F}} : \mathcal{F} \to \iota(a\mathcal{F})$ is the canonical map from a presheaf to its sheafification.

The diagonal functor $\Delta : \mathcal{C} \to \mathcal{C} \times \mathcal{C}$ sending $A \mapsto (A, A)$ has a right adjoint (when products exist): the product functor $\times : \mathcal{C} \times \mathcal{C} \to \mathcal{C}$. We have $\Delta \dashv \times$:

$\mathrm{Hom}((A,A), (B,C)) \cong \mathrm{Hom}(A, B \times C)$

For a multiplicative set $S \subseteq R$, the localization functor $S^{-1}(-) : R\text{-}\mathbf{Mod} \to S^{-1}R\text{-}\mathbf{Mod}$ is left adjoint to the restriction of scalars $S^{-1}R\text{-}\mathbf{Mod} \to R\text{-}\mathbf{Mod}$.

For a continuous map $f : X \to Y$, the inverse image functor $f^{-1} : \mathbf{Sh}(Y) \to \mathbf{Sh}(X)$ is left adjoint to the direct image $f_* : \mathbf{Sh}(X) \to \mathbf{Sh}(Y)$:

$\mathrm{Hom}_{\mathbf{Sh}(X)}(f^{-1}\mathcal{G}, \mathcal{F}) \cong \mathrm{Hom}_{\mathbf{Sh}(Y)}(\mathcal{G}, f_*\mathcal{F})$

An adjunction between poset categories (viewed as categories) is a Galois connection: $f(a) \leq b \iff a \leq g(b)$ for order-preserving maps $f : P \to Q$ and $g : Q \to P$. The classical Galois correspondence between subfields and subgroups is a Galois connection.

The Stone-Cech compactification functor $\beta : \mathbf{Top} \to \mathbf{CHaus}$ is left adjoint to the inclusion $\iota : \mathbf{CHaus} \hookrightarrow \mathbf{Top}$: $\mathrm{Hom}_{\mathbf{CHaus}}(\beta X, K) \cong \mathrm{Hom}_{\mathbf{Top}}(X, K)$ for compact Hausdorff $K$.

The forgetful functor $U : \mathbf{Top} \to \mathbf{Set}$ has both a left adjoint (discrete topology) and a right adjoint (indiscrete topology). This means $\mathrm{Disc} \dashv U \dashv \mathrm{Indisc}$.

In pointed topological spaces, the suspension functor $\Sigma$ is left adjoint to the loop space functor $\Omega$: $[\Sigma X, Y]_* \cong [X, \Omega Y]_*$. This adjunction is the foundation of stable homotopy theory.