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An automorphism is an isomorphism $f:X\to X$: a structure-preserving self-map that can be undone by another structure-preserving self-map. Equivalently, it is an invertible endomorphism. Its inverse satisfieswhere $1_X$ is the identity morphism. Both maps must be morphisms in the chosen category. For ordinary groups, rings, and vector spaces over a fixed field, a bijective morphism automatically has a compatible inverse. For topological spaces, an automorphism is a homeomorphism from the space to itself: a continuous bijection needs a continuous inverse as well. See Riehl, Category Theory in Context, Definition 1.1.10 and Example 1.1.11.

For example, $T(x,y)=(x+y,y)$ is an automorphism of the vector space ${\mathbb{R}}^2$ over the real numbers, with inverse $T^{-1}(u,v)=(u-v,v)$. Both are linear maps. The linear endomorphism $P(x,y)=(x,0)$ from endomorphism preserves addition and scalar multiplication too, but discards a coordinate and cannot be undone. Preserving the structure and preserving enough information for reversal are separate requirements.

The automorphisms of a fixed object in a locally small category form a group under composition in a category: the identity morphism is the identity element, and the inverse element of a composite reverses the order of its factors. This makes reversible internal changes calculable. For a group, the resulting automorphism group records which relabelings preserve all products; the examples in group automorphism show how restrictive that condition can be.

Given a pullback in a category $A=B{\times }_{D}C$ and a morphism $D^{\prime }\to D$, assume the indicated pullbacks in a category exist. The four base-changed objects form a pullback in a category. A compatible pair of morphisms into $B{\times }_D{D}^{\prime }$ and $C{\times }_D{D}^{\prime }$ gives morphisms into $B,C$ and a common morphism into $D^{\prime }$. The first two morphisms factor uniquely through $A$, and their common image in $D$ permits unique factorization through $A{\times }_D{D}^{\prime }$. This proves the displayed canonical isomorphism without an assumption that the whole category is complete.

A bifunctor is a functor on the product of two categories. It assigns $F(c,d)$ to each pair of objects and $F(u,v)$ to a pair of morphisms, preserving identity morphisms and composition in a category. Equivalently, it is functorial separately in each argument, with the two actions commuting: $F(1,v)F(u,1)=F(u,1)F(1,v)$ whenever the expressions have the corresponding sources and targets. The product category has pairs as objects, pairs of morphisms as arrows, and componentwise composition in a category. A monoidal tensor product is a bifunctor from a category times itself.

For small categories $\mathcal{C},\mathcal{D}$, a functor $F:\mathcal{C}\to \mathcal{D}$ and $Q:\mathcal{C}\to \mathbf{Set}$, choose an infinite cardinal number $\kappa$ bounding the number of morphisms of $\mathcal{D}$ and the total number of elements of $Q$. Given a natural transformation $\eta :Q\to GF$, generate a subfunctor $G^{\prime }\subseteq G$ from all ${\eta }_c(x)$ by applying every outgoing $\mathcal{D}$-morphism. Each $G^{\prime }(d)$ has at most $\kappa$ elements. Label bounded values by subsets of $\kappa$; only a set of such functors and transformations exists. These factorizations give the solution-set condition for precomposition, whose left adjoint is the left Kan extension.

Choose one regular functor witnessing the regular-logic separation of a proper subobject for each proper subobject represented by a monomorphism of the small regular category. Their product is regular and reflects invertibility of monomorphisms. If an arbitrary morphism becomes invertible, its image inclusion becomes invertible and so does the diagonal of its kernel pair. The original morphism is both a regular epimorphism and a monomorphism, hence invertible. The target is a power of sets as a capital regular category. This proves an isomorphism-reflecting regular representation without asserting full faithfulness.

Fix a monad on $\mathcal{C}$. Objects are adjunctions with left-hand category $\mathcal{C}$ whose induced monads are identified with the fixed monad. A morphism between two such adjunctions is a functor between their right-hand categories commuting with both adjoints and respecting the specified units and counits. Strict identifications give a category; coherent identifications give the analogous universal property up to compatible natural isomorphism.

The category whose objects are finite sets and whose morphisms are all functions between them. Finite categorical limits and colimits are the ordinary set constructions. It is essentially small: a skeleton of a category consists of the sets $\{0,\dots ,n-1\}$ for $n\in \mathbb{N}$. Its Yoneda embedding preserves limits but generally does not preserve coproducts: evaluating $y1⨿y1$ at a two-element set gives two elements, while evaluating $y2$ gives four.

Objects are small categories and morphisms are functors. Its products pair objects and arrows componentwise. Its exponential objects are functor categories, as shown by the currying adjunction for small categories.

Category theory studies mathematical structures through the morphisms between them. A category specifies which objects of a category and arrows are allowed; a functor translates these objects and arrows into another category; a natural transformation compares two such translations coherently. For example, the category of groups records both groups and group homomorphisms, so it remembers which maps respect multiplication.

The usefulness appears in constructions we already know. A Cartesian product combines two sets, while a direct product of groups combines two groups. Both satisfy the same universal property of a categorical product: any pair of maps into the factors comes from exactly one map into the product. This gives a reusable way to construct maps and establish isomorphisms, without depending on a particular choice of coordinates or element labels. In a partially ordered set regarded as a category, the corresponding categorical product is a greatest lower bound. The shared definition exposes a connection between pairing data and finding the best common bound.

Start with the computed examples under category, functor, natural transformation, and product in a category. Their arrow equations are practical tests: do both routes in a commutative diagram produce the same result? Later, the same language organizes tensor products, adjunctions, and the fundamental group of a topological space. Emily Riehl's textbook develops these basic notions before using them to compare constructions across different subjects. ^[ref]

In their 1945 paper, Samuel Eilenberg and Saunders Mac Lane set out the notions of category, functor, and natural transformation. The introduction uses the relation between a vector space and its dual space to ask which comparisons are natural, setting up a precise language for compatibility with maps. ^[ref]

A cogenerating set is a set-indexed family $(Q_i{)}_{i\in I}$ such that whenever $f\ne g:A\to B$, there is a morphism $q:B\to Q_i$ with $qf\ne qg$. It generalizes a single coseparator. In a locally small category with the needed products in a category, it gives an evaluation embedding into cogenerator products.

In an abelian category, the coimage of a morphism is its domain modulo its kernel in a category, expressed as a cokernel in a category. The canonical arrow from the coimage to the image is an isomorphism, giving the image factorization in an abelian category.

A morphism $f:C\to D$ satisfying ${\Delta }_{D}f=(f\otimes f){\Delta }_C$ and ${\epsilon }_{D}f={\epsilon }_C$.