A functor between two categories

A morphism between two functors

A bijection between two sets

A function between two groups

A function $\eta: \mathcal{C} \rightarrow \mathcal{D}$ such that $\eta(F(x)) = G(x)$ for all $x \in \mathcal{C}$

A function $\eta: \mathcal{C} \rightarrow \mathcal{D}$ such that $\eta(F(x)) = G(\eta(x))$ for all $x \in \mathcal{C}$

A function $\eta: \mathcal{C} \rightarrow \mathcal{D}$ such that $\eta(F(x)) = \eta(G(x))$ for all $x \in \mathcal{C}$

A function $\eta: \mathcal{C} \rightarrow \mathcal{D}$ such that $\eta(F(x)) = G(F(x))$ for all $x \in \mathcal{C}$

$F$ and $G$ are isomorphic functors

$F$ and $G$ are equivalent functors

$F$ and $G$ are naturally equivalent functors

$F$ and $G$ are adjoint functors

A natural transformation $\eta: F \rightarrow G$ that is an isomorphism

A natural transformation $\eta: F \rightarrow G$ that is a bijection

A natural transformation $\eta: F \rightarrow G$ that is a homeomorphism

A natural transformation $\eta: F \rightarrow G$ that is a diffeomorphism

A result in category theory that relates functors to natural transformations

A result in category theory that relates categories to functors

A result in category theory that relates functors to adjunctions

A result in category theory that relates categories to natural transformations

A condition on a natural transformation that ensures that it is an isomorphism

A condition on a natural transformation that ensures that it is a bijection

A condition on a natural transformation that ensures that it is a homeomorphism

A condition on a natural transformation that ensures that it is a diffeomorphism

A theorem that states that any two natural transformations between the same functors are equal

A theorem that states that any two natural transformations between the same functors are isomorphic

A theorem that states that any two natural transformations between the same functors are equivalent

A theorem that states that any two natural transformations between the same functors are naturally equivalent

A natural transformation between two natural transformations

A natural transformation between two functors

A natural transformation between two categories

A natural transformation between two adjunctions

A functor between two categories that is contravariant in one argument and covariant in the other

A functor between two categories that is covariant in both arguments

A functor between two categories that is contravariant in both arguments

A functor between two categories that is neither covariant nor contravariant in either argument

A natural transformation that satisfies the Beck-Chevalley condition

A natural transformation that satisfies the Yoneda lemma

A natural transformation that satisfies the coherence theorem

A natural transformation that does not satisfy any of the above conditions

A natural transformation that satisfies the Beck-Chevalley condition

A natural transformation that satisfies the Yoneda lemma

A natural transformation that satisfies the coherence theorem

A natural transformation that satisfies all of the above conditions

A natural transformation that is not strict

A natural transformation that is lax

A natural transformation that is dinatural

A natural transformation that is profunctorial

A natural transformation that is an isomorphism

A natural transformation that is a bijection

A natural transformation that is a homeomorphism

A natural transformation that is a diffeomorphism

A functor that extends a functor from a category to its category of presheaves

A functor that extends a functor from a category to its category of sheaves

A functor that extends a functor from a category to its category of toposes

A functor that extends a functor from a category to its category of groupoids

A functor that extends a functor from a category to its category of Kan complexes

A functor that extends a functor from a category to its category of simplicial complexes

A functor that extends a functor from a category to its category of CW complexes

A functor that extends a functor from a category to its category of topological spaces