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Given morphisms $f:A\to B$ and $g:B\to C$, their composite is $gf:A\to C$. Composition is associative and has the identity morphisms as units.

Two objects of a category lie in the same connected component if a finite zigzag of morphisms, with either orientation allowed, joins them. This is an equivalence relation; its classes give full subcategories, each a connected category. For a small category, the collection of components is a set denoted ${\pi }_0(\mathcal{C})$. A functor induces a function on components. Empty categories have no components.

An endomorphism is a morphism $f:X\to X$: its source and target are the same object, with the same specified structure. For a group it is an operation-preserving group homomorphism back into that group; for a vector space it is a linear endomorphism; for a ring it is a ring endomorphism. A self-map must satisfy the chosen compatibility conditions to qualify. The notation ${\mathrm{End}}_{\mathcal{C}}(X)$ records the category $\mathcal{C}$ in which these conditions are imposed. See Riehl, Category Theory in Context, Definition 1.1.10 and the following paragraph.

An endomorphism can lose information or miss part of its target. On the additive integers, $d(k)=2k$ preserves addition and is injective, but is not surjective: no integer maps to $1$. On the vector space ${\mathbb{R}}^2$ over the real numbers, the linear endomorphism $P(x,y)=(x,0)$ loses the second coordinate, so it is neither injective nor surjective. Neither is an automorphism. The same doubling rule becomes a group isomorphism when its target is the additive even integers, as explained under isomorphism.

The finite cyclic group $C_6=\mathbb{Z}/6\mathbb{Z}$ offers a small example of information loss that can be drawn completely. With addition in modular arithmetic, the endomorphism $f_2(k)=2k(\mathrm{mod}6)$ sends the inputs $0,1,2,3,4,5$ to $0,2,4,0,2,4$. Its image of a group homomorphism is $\{0,2,4\}$ and its kernel of a group homomorphism is $\{0,3\}$. Each reached output has two inputs; the other three outputs are missed.

For an endomorphism of a group or ring with a finite underlying set, injectivity is equivalent to surjectivity by counting; either condition makes it an automorphism. The additive integers example shows why the finiteness assumption matters. A linear endomorphism of a finite-dimensional vector space also has equivalent injectivity and surjectivity, this time by the rank-nullity theorem even when its underlying set is infinite.

Self-maps can be iterated without changing the object: $f^2=f\circ f$, $f^3=f\circ f\circ f$, and so on. Composition in a category and the identity morphism make the endomorphisms of a fixed object in a locally small category into a monoid. This packages repeated compatible operations, including irreversible ones, into a single structure.

The general language has a documented historical milestone in the 1945 paper of Samuel Eilenberg and Saunders Mac Lane. They defined categories of objects and compatible maps, and distinguished maps admitting inverses within that category. Their examples included groups, vector spaces, and topological spaces. This gave a common framework for compatible self-maps and their invertible cases. See Eilenberg and Mac Lane, General Theory of Natural Equivalences, pp. 234 and 237–239.

For a small cogenerating family $(Q_i)$ in a locally small category, the evaluation arrow $C\to {\prod }_{i,q\in \mathcal{C}(C,Q_i)}{Q}_i$ is a monomorphism whenever that small product in a category exists. Equality after all its projections is equality after every map to a cogenerator, hence equality of the original parallel morphisms.

The braided monoidal category generated by one object: objects are parenthesized tensor expressions in that object and the unit, and morphisms are structural isomorphisms and crossings subject to precisely the monoidal category and braiding axioms.

A monoid object $A$ with a morphism $\epsilon :A\to I$ such that $\epsilon m$ and a suitable coevaluation make $(A,A)$ a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

If each standard counit presentation is a coequalizer, the Eilenberg-Moore comparison functor is full and faithful. An monad algebra morphism $\alpha :GB\to GC$ makes ${\epsilon }_{C}F\alpha$ equalize $FG{\epsilon }_B$ and ${\epsilon }_{FGB}$; its unique descent is the required morphism $B\to C$. The triangle identities for an adjunction make $G{\epsilon }_B$ a split epimorphism, so descent has underlying arrow exactly $\alpha$. Epimorphic cancellation at ${\epsilon }_B$ proves faithfulness.

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

A generating set is a set-indexed family $(P_i)$ such that unequal parallel morphisms $f,g:A\to B$ are distinguished by some $h:P_i\to A$, with $fh\ne gh$. It is dual to a cogenerating set.

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 morphism in an abelian category, its coimage $\mathrm{coker}(\ker f)$ is canonically isomorphic to its image $\ker (\mathrm{coker}f)$. This yields an epimorphism followed by a monomorphism, unique up to unique compatible isomorphism. Every epi-mono factorization has this same middle object.

An isogeny is a surjective morphism of algebraic groups with finite kernel. For abelian varieties it is a finite surjective group morphism. An isogeny of elliptic curves is its one-dimensional abelian-variety specialization.

An isomorphism is a morphism with a two-sided inverse that is also a morphism. Two objects are isomorphic when such a map exists between them. For groups, rings, and vector spaces, a bijective structure-preserving map has a structure-preserving inverse; in other settings the inverse condition needs separate attention, as for a homeomorphism between topological spaces.

For example, $f(k)=2k$ is a group isomorphism from the additive integers to the additive even integers. It is bijective onto that target, satisfies $f(a+b)=f(a)+f(b)$, and has inverse $2k\mapsto k$. The two groups have the same addition pattern even though they are different sets. It is not a ring isomorphism for their inherited multiplications, since $f(ab)=2ab$ while $f(a)f(b)=4ab$.

A morphism of monads on one category is a natural transformation $\alpha :S\Rightarrow T$ with $\alpha {\eta }^S={\eta }^T$ and $\alpha {\mu }^S={\mu }^T(T\alpha )(\alpha S)$. These equations preserve the unit and multiplication. A terminal endofunctor in a full composition-closed subcategory containing the identity receives a unique such morphism from every monad whose functor belongs to that subcategory.

A functor $F:\mathcal{C}\to \mathbf{Set}$ is a monofunctor when $F(f)$ is an injective function for every morphism $f$ of $\mathcal{C}$.

A morphism $f:A\to B$ satisfying $f{m}_A=m_B(f\otimes f)$ and $f{j}_A=j_B$.

For a natural transformation $\theta :F\Rightarrow G$, naturality is the equation $G(f){\theta }_A={\theta }_{B}F(f)$ for each morphism $f:A\to B$. Thus transforming and then applying a functor gives the same result as applying that functor and then transforming. It makes the component maps a coherent transformation of functors.

The objects are the entities between which a category specifies morphisms. Each object has an identity morphism.