structural:summarization
Diagrams as functors

A commutative diagram in a category C can be interpreted as a functor from an index category J to C; one calls the functor a diagram.

More formally, a commutative diagram is a visualization of a diagram indexed by a poset category. Such a diagram typically includes:

Conversely, given a commutative diagram, it defines a poset category, where:

However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram). As a simple example, the diagram of a single object with an endomorphism (f: X → X {\displaystyle f\colon X\to X}), or with two parallel arrows (∙ ⇉ ∙ {\displaystyle \bullet \rightrightarrows \bullet }, that is, f, g: X → Y {\displaystyle f,g\colon X\to Y}, sometimes called the free quiver), as used in the definition of equalizer need not commute. Further, diagrams may be messy or impossible to draw, when the number of objects or morphisms is large (or even infinite).