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For a locally small category $\mathcal{C}$, an object $C$, and a categorical presheaf $X:{\mathcal{C}}^{\mathrm{op}}\to \mathbf{Set}$, the Yoneda lemma gives the natural bijectionIts two maps are explicitlyFor $u:A^{\prime }\to A$, the equation $X(u)X(f)(x)=X(fu)(x)$ proves naturality of $\Psi (x)$. Evaluating it at the identity morphism gives $\Phi \Psi (x)=x$. Conversely, naturality of $\alpha$ at $f:A\to C$ givesso $\Psi \Phi (\alpha )=\alpha$. Evaluation at the identity and transport of an element along a morphism are mutually inverse.

The bijection is natural in both variables: a natural transformation $\theta :X\Rightarrow Y$ sends $\Phi (\alpha )$ to ${\theta }_C(\Phi (\alpha ))$, matching $\Phi (\theta \alpha )$; and $g:C\to C^{\prime }$ givesFor completeness, the covariant Yoneda lemma for $Y:\mathcal{C}\to \mathbf{Set}$ is $\mathrm{Nat}(\mathcal{C}(C,-),Y)\cong Y(C)$, with $\alpha \mapsto {\alpha }_C(1_C)$ and inverse $y\mapsto (f:C\to A\mapsto Y(f)(y))$.

The representing object in the functor category $[J,\mathcal{C}]$ is the constant diagram in a category ${\Delta }_{J}C$. If $p_j:{\lim }_j{E}_j\to E_j$ is the chosen categorical limit cone, the natural bijection isA natural transformation from the constant diagram in a category is precisely a categorical cone with vertex $C$, and its inverse is the unique mediating morphism from the universal property of the categorical limit. For $\alpha :E\Rightarrow E^{\prime }$, the defining equations for $\lim \alpha$ show that postcomposition by $\lim \alpha$ corresponds to postcomposition by $\alpha$ on the right. This establishes naturality in $E$ and proves representability by the constant diagram.

Use the orientation $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$. An adjunction $F⊣G$ is equivalently specified by the natural transformationscalled the unit and counit of an adjunction, satisfying the triangle identities for an adjunctionThe corresponding natural bijection is $\mathcal{D}(FC,D)\cong \mathcal{C}(C,GD)$, with $f\mapsto G(f){\eta }_C$ and inverse $g\mapsto {\epsilon }_{D}F(g)$. The two triangular equations are the required compatibility conditions. No proof of equivalence of the formulations is needed here.

The right-adjoint criterion using comma-category colimits is as follows. Under the given colimit-preservation hypothesis, with preservation including the possibly large colimits below, $F:\mathcal{X}\to \mathcal{A}$ has a right adjoint exactly when, for every $A$, the projectionhas a colimit in $\mathcal{X}$. The right adjoint is obtained from these comma-category colimits. Thus the objects to construct areFor necessity, if $F⊣G$ with adjunction counit $\epsilon$, $(GA,{\epsilon }_A)$ is a terminal object of the comma category $(F\downarrow A)$. Its unique incoming morphisms give a colimit cocone for $U_A$, exactly as for the elements projection in the preceding part.

For sufficiency, choose such a colimit $L$ with legs $u_{X,f}:X\to L$. The morphisms $f:FX\to A$ are a cocone on $F{U}_A$. Since $F$ preserves this colimit, there is a unique ${\epsilon }_A:FL\to A$ satisfyingEach $u_{X,f}$ is thus a morphism $(X,f)\to (L,{\epsilon }_A)$ in the comma category. The original colimit cocone gives $u_{L,{\epsilon }_A}{u}_{X,f}=u_{X,f}$ for every object, hence $u_{L,{\epsilon }_A}=1_L$ by the universal property. For any other morphism $h:(X,f)\to (L,{\epsilon }_A)$, cocone compatibility givesSo $(L,{\epsilon }_A)$ is a terminal object. Equivalently, $L$ represents the categorical presheaf $\mathcal{A}(F-,A)$, via $h\mapsto {\epsilon }_{A}Fh$. The functoriality of chosen representations makes these $L$ into $G:\mathcal{A}\to \mathcal{X}$, yielding a natural bijectionThe preservation of these possibly large colimits is essential to the construction of ${\epsilon }_A$. Under a small-only interpretation, the preceding ordinal counterexample also disproves the unqualified converse here. Take $F=X^{\mathrm{op}}:\mathcal{C}\to {\mathbf{Set}}^{\mathrm{op}}$. It preserves all small colimits. For a nonempty set $A$, the comma category $(F\downarrow A)$ has one object over each ordinal and none over $\infty$, so its projection has colimit $\infty$. For $A=\varnothing$, the projection is the identity of $\mathcal{C}$, again with colimit $\infty$. Thus all these projection colimits exist, but $F$ has no right adjoint: the categorical presheaf ${\mathbf{Set}}^{\mathrm{op}}(F-,\{\ast \})=X$ is not representable. This makes the large-preservation qualification substantive.

For the monad $\mathbb{T}=(T,\eta ,\mu )$ define the free algebra functor byThe unit and associativity identities of the monad make $(TX,{\mu }_X)$ an algebra for a monad, and naturality of $\mu$ makes $Tf$ a morphism of algebras for a monad. Let $G^{\mathbb{T}}$ be the forgetful functor. For $(A,a)$ in the Eilenberg-Moore category, setThe inverse candidate is a monad algebra morphism, sinceThe two composites are identities:These use respectively the unit law for a monad algebra, the algebra-morphism equation, and a unit identity of the monad. The formulas commute with precomposition in $X$ and postcomposition by monad algebra morphisms, so they form a natural bijection. Therefore the free-algebra functor is left adjoint to forgetting:Its adjunction unit is ${\eta }_X$, and its adjunction counit at $(A,a)$ is $a:(TA,{\mu }_A)\to (A,a)$.