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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.

For a small category with finite limits, a set-valued functor preserving them has a category of elements with finite limits, so its opposite is a filtered category. The covariant density presentation is therefore a filtered colimit of representable functors. Conversely, representable functors preserve categorical limits, and filtered colimits commute with finite limits in sets, proving the criterion.

A functor discarding some structure, for example an action on a module. It need not be full or faithful in general.

For a monad $(T,\eta ,\mu )$, the free algebra functor $F:\mathcal{C}\to {\mathcal{C}}^T$ sends $X$ to $(TX,{\mu }_X)$ and $f$ to $Tf$. It is a left adjoint to the forgetful functor $U$, via $h\mapsto h{\eta }_X$ and $u\mapsto \gamma Tu$ between ${\mathcal{C}}^T(FX,(C,\gamma ))$ and $\mathcal{C}(X,C)$.

Take two distinct objects and one nonidentity arrow for each element of $S$, without arrows between different copies. A functor from this category to $\mathcal{C}$ is exactly a choice of one arbitrary arrow of $\mathcal{C}$ for each element of $S$. Consequently this construction is left adjoint to the functor taking the set of all arrows of a small category. Its free object has three arrows per generator, including the two identity arrows.

For a functor $U:\mathcal{C}\to \mathcal{D}$ between locally small categories, with $\mathcal{C}$ a complete category, $U$ has a left adjoint if and only if it preserves small categorical limits and satisfies the solution-set condition. The dual statement uses cocompleteness, small-colimit preservation and weakly terminal solution sets to characterize existence of a right adjoint.

A lax monoidal functor $\phi$ and opmonoidal functor $\psi$ on the same underlying functor, satisfying ${\psi }_{A\otimes B,C}{\phi }_{A,B\otimes C}=({\phi }_{A,B}\otimes 1)(1\otimes {\psi }_{B,C})$ and ${\psi }_{A,B\otimes C}{\phi }_{A\otimes B,C}=(1\otimes {\phi }_{B,C})({\psi }_{A,B}\otimes 1)$ in coherent notation. It transports dual pairs in a monoidal category and Frobenius monoids.

A functor $F:\mathcal{C}\to \mathcal{D}$ is full when each function $\mathcal{C}(A,B)\to \mathcal{D}(FA,FB)$ is surjective. It is a faithful functor when each is injective. A full and faithful functor identifies its source, up to equivalence, with a full subcategory of the target. Fullness only concerns morphisms between image objects; it does not require every target object to lie in the image.

A full subcategory contains all ambient morphisms between its chosen objects. Its inclusion is a fully faithful functor.

A commuting square $vf=f^{\prime }u$ induces a unique map between their images by $I(u,v)p=p^{\prime }u$ and $i^{\prime }I(u,v)=vi$. The cokernel in a category property supplies existence, and cancellation of the epimorphic $p$ supplies uniqueness. These equations prove identity and composition laws, giving a functor from the arrow category to the abelian category.

If $H:{\mathcal{A}}^{\mathrm{op}}\times \mathcal{B}\to \mathbf{Set}$ has chosen representations ${\psi }_B:\mathcal{A}(-,GB)\cong H(-,B)$, the Yoneda lemma determines a unique functor $G$ on morphisms by $\mathcal{A}(-,Gb)={\psi }_{B^{\prime }}^{-1}H(-,b){\psi }_B$. This makes the representing natural isomorphisms natural in $B$.

Objects are geometric formulas in context modulo provable equivalence. Morphisms are provably total, single-valued geometric relations; composition is existential conjunction. Conjunction and equality give finite limits. Interpreting formulas in a model gives a finite-limit-preserving functor, with the additional continuity condition supplied by the geometric syntactic topology.

A groupoid is a category in which every morphism is invertible. A group is a groupoid with one object, while an equivalence relation defines a groupoid with exactly one arrow between any related pair of objects. Every functor sends invertible arrows to invertible arrows.

For a projective scheme $X$ over a field $k$, the Hilbert functor assigns to a $k$-scheme $T$ the set of closed subschemes $W\subseteq X{\times }_{k}T$ which are of finite presentation, proper and flat over $T$. Pullback of families makes it a contravariant functor. After choosing a very ample line bundle, requiring a fixed fibre Hilbert polynomial $P$ gives the subfunctor represented by ${\mathrm{Hilb}}^P(X)$. The unrestricted functor is represented by the disjoint union of these Hilbert schemes.

A categorical cone $L\to E_i$ is a categorical limit precisely when $\mathcal{C}(C,L)\to {\lim }_i\mathcal{C}(C,E_i)$ is a bijection for every object $C$. This is exactly the universal property expressed as a test on hom-sets. Consequently a functor preserves existing categorical limits if every composite with a covariant representable functor does.

In the positive finite-limit convention, a Horn theory uses atomic finite-conjunction sequents; uniquely witnessed existential extensions remain Cartesian. Its models are the finite-limit-preserving functors out of its Cartesian syntactic category. This convention does not insert arbitrary existential witnesses or falsity as Cartesian constructors. Its classifier is the presheaf classifier of a Horn theory.

The constant singleton functor must be a retract in a category of a covariant representable functor. Necessity follows by applying a colimit-preserving global-sections functor to the canonical coproduct of representables onto the terminal functor. Conversely its Hom functor is a retract of evaluation, preserves colimits and has a right adjoint. Such a retract is represented by an initial object in the idempotent completion. For an idempotent-complete category an actual initial object suffices; without this hypothesis an absorbing-zero monoid gives a counterexample to the unqualified claim.