Universality of a concept here means wide conceptual and practical usefulness in mathematics and applications. The function concept owes its universality to simplicity, generality and powerful algebraic properties. Advantages proven in the sciences at large significantly benefit computing science as well. Universality critically depends on the definitional choices. The first half of this paper shows that a “function” in the sense prevalent throughout the sciences, namely, as fully specified by its domain and its values, entails the characteristics that most contribute to universality. This link is clarified by some less well-understood aspects, including the role of function types as partial specifications, the ramifications of having composition defined for any pair of functions, and unification by capturing various notions not commonly seen as functions. Simple but representative examples are given in diverse areas, mostly computing. When a codomain appears at all in basic textbooks, it mostly involves a self-contradicting definition, corrected by the labeled function variant. Either way, it severely reduces universality, especially for composition. Yet, the axiomatization of category theory common in theoretical computing science offers no other choice. The second half explores how waiving one axiom generalizes category theory to include a wider variety of concepts, primarily the conventional function variant. It is also shown how this can be done unobtrusively for typical categorical notions, such as products, coproducts, functors, natural transformations, adjunctions, Galois connections, and auxiliary concepts, illustrated by example definitions and technical comments. Allowing the familiar function variant renders category theory more appealing to a wider group of scientists. A lesson for mathematics in general is Rogaway’s maxim: “Your definitional choices should be justified”!

这里概念的普遍性意味着数学和应用中广泛的概念和实际用途。函数概念的普遍性归因于简单性、普遍性和强大的代数性质。整个科学领域所证明的优势也极大地有利于计算科学。普遍性很大程度上取决于定义的选择。本文的前半部分表明，在整个科学中普遍存在的意义上的“功能”，即由其领域和价值完全指定的“功能”，包含了最有助于普遍性的特征。这种联系通过一些不太容易理解的方面得到了澄清，包括函数类型作为部分规范的作用、为任何函数对定义组合的后果，以及通过捕获通常不被视为函数的各种概念来进行统一。不同领域（主要是计算领域）给出了简单但具有代表性的示例。当共域出现在基础教科书中时，它主要涉及一个自相矛盾的定义，并通过标记的函数变体进行纠正。不管怎样，它都会严重降低通用性，尤其是对于组合而言。然而，理论计算科学中常见的范畴论公理化没有提供其他选择。后半部分探讨了如何放弃一个公理来概括范畴论以包括更广泛的概念，主要是传统的函数变体。它还展示了如何对于典型的分类概念（例如乘积、余积、函子、自然变换、附加词、伽罗瓦连接和辅助概念）不引人注意地完成此操作，并通过示例定义和技术注释进行说明。允许熟悉的函数变体使得范畴论对更广泛的科学家群体更有吸引力。罗格威的格言是数学的一课：“你的定义选择应该是合理的”！