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A comonoid morphism $f:H\to K$ induces a functor $f_{\ast }$ on right comodules, replacing $\rho$ by $(1\otimes f)\rho$ and preserving underlying objects and arrows. For bimonoids, it is strict monoidal exactly when $f$ is also a monoid morphism.

A coseparator $S$ in a category distinguishes parallel arrows by postcomposition: if $u\ne v:X\to B$, some $t:B\to S$ satisfies $tu\ne tv$. Equivalently, the contravariant functor $\mathcal{C}(-,S)$ is faithful. In a complete category that is a locally small category, the evaluation arrow $B\to {\prod }_{t\in \mathcal{C}(B,S)}S$ is a monomorphism.

For a set-valued covariant functor on a small category, the Yoneda lemma sends the element $x\in F(c)$ to the transformation $\mathcal{C}(c,-)\to F$. These transformations form a cocone indexed by the opposite category of elements. At $a$, send the representative $(c,x,u:c\to a)$ to $F(u)x$. It is surjective using $(a,z,1_a)$; two representatives of $z$ are identified through that same object of the category of elements. This proves the displayed presentation pointwise.

A functor $\mathcal{D}\times \mathcal{C}\to \mathcal{E}$ is uniquely a functor $\mathcal{D}\to [\mathcal{C},\mathcal{E}]$: fix its first coordinate and use its first-coordinate arrows as natural transformations. Uncurrying evaluates those transformations and second-coordinate arrows. The adjunction unit inserts a fixed first coordinate, and the adjunction counit is evaluation. This makes the category of small categories cartesian closed.

Geometric morphisms into a sheaf topos correspond to continuous flat functors from its site into the domain topos, with continuity meaning that covers become jointly epimorphic families. Pulling back sheafified representables gives the functor; the tensor construction gives the inverse image in the other direction. This representation theorem should not be confused with the separately named Diaconescu theorem about choice and excluded middle.

A functor with a unique lift, from each specified object $c$, of every arrow $Fc\to d$. The lifted arrow's target is part of the uniqueness requirement. This is dual to a discrete fibration. Lifting a cocone of a finite connected category gives a cocone upstairs: lifts of its legs have a common target because uniqueness identifies targets along every diagram arrow. Thus a discrete opfibration over a weakly filtered category has weakly filtered domain, including the possibility of an empty domain.

An endofunctor is a functor from a category to itself. A monad adds a unit and multiplication to an endofunctor, satisfying the unit and associativity laws.

If $\mathcal{C}$ has finite categorical limits, every functor $F:\mathcal{C}\to \mathcal{D}$ preserving pullbacks in a category also preserves equalizers, even when it does not preserve a terminal object. Factor it through the slice category $\mathcal{D}/F1$ by sending $A$ to $F(A\to 1)$. This lifted functor preserves pullbacks because the slice forgetful functor reflects them; it also preserves the terminal object, so it preserves finite limits. The slice forgetful functor preserves equalizers, completing the proof.

In a category with equalizers, the equalizer of ${\eta }_{TA},T{\eta }_A$ defines a functor $R$ by ${\beta }_{B}Rf=Tf{\beta }_A$. It has unit $\alpha$ and multiplication $\theta$ characterized byCancelling the monomorphisms ${\beta }_A$ reduces its unit and associativity laws to those of $T$, so $\beta :R\Rightarrow T$ is a monad morphism. The split equalizers associated with a monad show that ${\alpha }_{TA}$ is invertible. If every $T{\beta }_A$ is monic, then every $T{\alpha }_A$ is invertible, the naturality square of $\beta$ at ${\beta }_A$ is a pullback in a category, and $R$ is an idempotent monad. This monicity hypothesis is essential to this last argument; arbitrary functors need not preserve monomorphisms.

An equivalence consists of functors $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ with natural transformation isomorphisms $GF\cong 1_{\mathcal{C}}$ and $FG\cong 1_{\mathcal{D}}$. A full and faithful functor which is essentially surjective, meaning every target object is isomorphic to an image object, is part of an equivalence under the usual choice convention. An isomorphism of categories is stronger: it has a strictly inverse functor.

A geometric morphism is essential when its inverse image has an additional left adjoint. For a functor of small categories, precomposition on presheaves has left Kan extension and Right Kan extension as its two adjoints. Precomposition preserves finite limits pointwise, so this is an essential geometric morphism.

A functor $F:\mathcal{C}\to \mathcal{D}$ is essentially surjective when every object of $\mathcal{D}$ is isomorphic to $FA$ for some $A\in \mathcal{C}$. Together with full and faithful, this characterizes an equivalence of categories under the appropriate axiom of choice convention. For large categories, choosing a quasi-inverse on all objects requires a universe or a class-choice convention.

The finality property of a functor means every required comma category is nonempty and connected. It is the condition defining a final functor; the displayed criterion uses categorical connected components.

If $\mathcal{C}$ has finite colimits and a monad $T$ preserves reflexive coequalizers, its Eilenberg-Moore category has finite colimits. The forgetful functor creates reflexive coequalizers preserved by $T$. The coproduct presentation for monad algebras yields binary coproducts, and $F0$ is initial. Any pair $a,b:X\rightrightarrows Y$ can then be replaced by the reflexive pair $[a,1],[b,1]:X+Y\rightrightarrows Y$, with identical coequalizers. Finite coproducts and coequalizers give all finite colimits.

A functor $F:\mathcal{C}\to \mathbf{Set}$ is finite-limit-preserving if it carries each finite limit cone to a limiting cone. When $\mathcal{C}$ is a small category with finite limits, this is equivalent to $F$ being a filtered colimit in a category of covariant representable functors, and to the opposite of its category of elements being a filtered category. It is also equivalent to every comma category $(S\downarrow F)$ having finite limits, for all sets $S$.

For a nonempty topological space $S$, the constant diagram functor on its open-set presheaf category lies in this five-term adjunction chain. The leftmost functor sends $A$ to the categorical presheaf equal to $A$ at the empty open set and empty elsewhere. The rightmost sends $A$ to the presheaf equal to $A$ at $S$ and a singleton elsewhere. A natural transformation from the former is determined by its empty-open component; a transformation into the latter is determined by its whole-space component. The middle adjunctions follow from restriction maps to the empty open set and from the whole space. Neither endpoint extends: the leftmost functor fails to preserve the terminal object, and the rightmost fails to preserve the initial object.

A fixed point of a group action is an element fixed by every group element. The set $A^G=\{a:g\cdot a=a\text{ for all }g\in G\}$ is the categorical limit of the action viewed as a functor from the one-object category associated with $G$ to the Category of sets.

A covariant functor into a Grothendieck topos is flat when its tensor extension from presheaves preserves finite limits. For set values, its category of elements with arrows carrying source elements to target elements is cofiltered; equivalently the functor is a filtered colimit of covariant representables. If the indexing category has finite limits, flatness is equivalent to preservation of finite limits. For a site, cover-to-joint-epimorphism continuity adds the condition needed by the Diaconescu equivalence for geometric morphisms.