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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.

In a balanced category with pullbacks in a category, pull a monomorphism back across the bottom map of a lifting square whose left map is an epimorphism. The resulting monic projection is epic because its composite with the lifted top map is epic. Balancedness makes that projection invertible, producing the unique lift. Conversely, if all monomorphisms are strong monomorphisms, applying the lifting property of a bimorphism to its own square gives its inverse, so the category is balanced. Pullback stability of epimorphisms is not needed.

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.

In a category with binary products in a category, a map with the left lifting property against monomorphisms is an epimorphism. If $ue=ve$, lift the square with right side $\Delta$ and bottom side $⟨u,v⟩$. The two projections of its filler give $u=v$. This argument does not require equalizers.

A capital regular category, in the convention of Definitions 1.1–1.3, is a regular category whose well-supported objects are well-pointed objects in a category. It does not require arbitrary objects to be detected by global points. A partially ordered set with finite meets, regarded as a category, is capital: its only well-supported object is the top element. Such a category can have many proper subterminal objects, all with no points. This example distinguishes the definition from the stronger convention that the global-sections functor reflects every isomorphism.

In the finite-limit convention, a Cartesian category is a category with all finite limits, equivalently a terminal object and all pullbacks in a category. Some authors use the term for finite products alone, so the convention matters in constructions requiring equalizers.

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.

This category has commutative monoids and their homomorphisms. Pointwise addition makes each hom-set a commutative monoid, and composition is additive in both variables. Finite cartesian products are also coproducts, using the coordinate injections with zero in the other coordinates and summing images. Therefore it is a semi-additive category, but it is not an additive category. The reflexive pair from the submonoid $\{(m,n):m\le n\le 2m\}$ of ${\mathbb{N}}^2$ to $\mathbb{N}$, using the two projections and section $m\mapsto (m,m)$, has an arrow from $1$ to $2$ and none from $2$ to $1$ in its represented graph. Thus a reflexive pair here need not yield a groupoid.

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.

The category of topological manifolds has topological manifolds without boundary as its objects of a category and all continuous maps between them as its categorical morphisms. Allowing every dimension of a manifold lets maps connect spaces of different dimensions. Identity functions are continuous, and function composition preserves continuity, so the category laws are inherited from functions. Its isomorphisms are exactly homeomorphisms.

In coordinate charts $(U,\phi )$ on $M$ and $(V,\psi )$ on $N$, a continuous map $f:M\to N$ is represented locally by $\psi \circ f\circ {\phi }^{-1}$ on $\phi (U\cap f^{-1}(V))$. Changing coordinate charts changes this expression through topological transition maps, while $f$ itself stays fixed. This viewpoint separates the underlying continuous map from the coordinates used to describe it; restricting to smooth manifolds and smooth maps gives the corresponding smooth category.

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]