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Use the preservation obstructions to extending an adjoint chain: a functor having a further left adjoint must preserve categorical limits, while one having a further right adjoint must preserve colimits.

At the left end of the presheaf category chain, $\Lambda$ does not preserve the terminal object. The terminal categorical presheaf has singleton value everywhere, whereas $(\Lambda 1)(S)=\varnothing$ because $S\ne \varnothing$. Thus $\Lambda$ cannot itself be a right adjoint. At the right end, $\nabla$ does not preserve the initial object: $(\nabla \varnothing )(\varnothing )=\{\ast \}$ since the empty open set is proper. Thus $\nabla$ cannot itself be a left adjoint.

For the category of small categories chain, take two functors from the terminal category $1$ to the two-object indiscrete category $I\{0,1\}$, selecting its distinct objects. Their equalizer is the empty category. But their images under $C={\pi }_0$ are the same function between singleton sets, whose equalizer is a singleton. Hence $C$ does not preserve equalizers and has no further left adjoint. Finally, $I1⨿I1$ is the two-object discrete category, while $I(1⨿1)$ has arrows between its distinct objects. The canonical comparison is not an isomorphism of categories, so $I$ fails to preserve this coproduct and has no further right adjoint.

Let $x$ denote the generating object, the image of the unique object of the terminal category. The free braided monoidal category on one object can be described syntactically. Its objects are all fully parenthesized expressions built from $x$, the monoidal unit object $I$, and a binary monoidal tensor product. Thus $x\otimes (I\otimes x)$ and $(x\otimes I)\otimes x$ are distinct objects, though canonically isomorphic. Its morphisms are generated by the associators, unitors, braidings and their inverses, closed under composition and tensor product, subject to the pentagon, triangle, naturality and hexagon axioms. No symmetry relation is imposed on the braiding. The embedding of the terminal category selects $x$.

The braid category $\mathbf{B}$ has objects the nonnegative integers, withwhere $B_n$ is the braid group on $n$ strands and $B_0,B_1$ are trivial. Composition is stacking braids, with the first morphism followed by the second. The monoidal tensor product is addition on objects and side-by-side juxtaposition of braids; its monoidal unit object is $0$. The associators and unitors are identities, so it is a strict monoidal category.

Choose the positive crossing convention once and for all. Its braiding $c_{m,n}:m+n\to n+m$ is the block braid moving the first $m$ strands over the next $n$ strands while retaining the order within each block. In particular $c_{1,1}={\sigma }_1$. The usual braid group relations express the naturality and hexagon laws for these block braids. The generating functor $1\to \mathbf{B}$ selects $1$.

The distinction is parenthesized tensor expressions versus strand counts: the first construction keeps the structural isomorphisms visible, while $\mathbf{B}$ has strict tensor arithmetic. These descriptions give the two requested free constructions without needing to establish their universal properties.

The slice over $A$ has morphisms $x:X\to A$ as objects and commuting triangles $yu=x$ as arrows $u:(X,x)\to (Y,y)$. Its terminal object is $1_A$, regardless of whether $\mathcal{C}$ has a terminal object. It is the comma category $(1_{\mathcal{C}}\downarrow A)$, viewing $A$ as a functor from the terminal category.

A functor between small categories is a strong epimorphism exactly when its image generates its codomain as a subcategory. A monomorphism between small categories is injective on objects and faithful, so its image can be identified with a subcategory. In a lifting square, all image generators lift to that subcategory; their composites lift too, and faithfulness forces their relations to agree. This gives the diagonal filler. Conversely, factoring through the generated image and applying the lifting property makes that inclusion an isomorphism.

A regular epimorphism need not be a full functor. For example, the unique functor from a two-object discrete category to the terminal category is a coequalizer of the two functors selecting its objects. A functor equalizing those selections has exactly one choice of factorization through the terminal category. Nevertheless the original functor is not full: there is no arrow between the two distinct source objects, whereas the target has its identity arrow.

Every regular epimorphism is strong, but the converse fails. The nonregular extremal epimorphism in the category of categories gives an image-generating functor for which imposing only its kernel pair identifications does not impose all target relations. Since small categories have pullbacks in a category, a regular epimorphism would have to be the coequalizer of that kernel pair.