Categories, morphisms, and composition

For objects $A$ and $B$, write $\mathcal C(A,B)$ for the collection of arrows with source $A$ and target $B$. Given $f:A\to B$ and $g:B\to C$, composition produces $g\circ f:A\to C$. The order of the symbols matters: the rightmost arrow acts first. Composition is not defined for arbitrary pairs of arrows. In a category of functions, an output set must match the next input set; in an abstract category this matching is part of the data.

Every object has an identity arrow, and composition must satisfy the identity and associativity laws. Associativity compares two bracketings of the same three-arrow path; it does not say that two arrows commute. Once these laws hold, longer composable paths have an unambiguous composite. A directed graph with no chosen composites is not yet a category. Its free category adds all finite paths, including empty paths as identities, and uses concatenation for composition.

A partially ordered set becomes a category by putting exactly one arrow $x\to y$ when $x\le y$. Reflexivity supplies identities and transitivity supplies composites. Because each relevant arrow is unique, the category laws follow automatically. The Hasse diagram usually suppresses identities and transitive arrows, so its visible edges are not the full morphism collection. Recovering the missing arrows is essential before counting maps or testing a universal property.

Another useful example has one object and a monoid of endomorphisms. The monoid multiplication is composition, and its unit is the object's identity. A group gives a special case in which every arrow is invertible. In a poset category, an isomorphism between $x$ and $y$ would require both inequalities, so antisymmetry forces equality. A preorder can instead have distinct isomorphic objects. Thus the relationship between equality and isomorphism depends on the chosen category.

A category is small when its objects and all its arrows form sets. It is locally small when each collection $\mathcal C(A,B)$ is a set, even if the objects form a proper class. Throughout, Set means sets in a fixed universe; it is large relative to that universe but locally small. Our finite categories are small. Later constructions involving categories of functors will state smallness assumptions rather than quietly treating every collection as a set.

An isomorphism $f:A\to B$ has an arrow $g:B\to A$ whose two composites are the appropriate identities. Both conditions are required. The inverse is unique: any other candidate $h$ satisfies $h=h(fg)=(hf)g=g$. Isomorphic objects can still have different names or different underlying sets. When we later call a construction unique up to unique isomorphism, uniqueness refers to an isomorphism compatible with the specified structure, not to every bare isomorphism between the objects.