In the field of category theory, the term "functorial" is used to describe a property or behavior of functors. A functor is a mapping between categories, which preserves the structure and relationships between objects and morphisms. When we say that something is functorial, it means that it can be extended or lifted to a functor in a natural way. This includes properties such as composition, identity preservation, and compatibility with morphisms. For example, if we have two categories A and B, and we define a functor F from A to B, we say that F is functorial if it respects the composition of morphisms in A and B. This means that for any two composable morphisms in A, their corresponding images under F will also be composable in B, and the image of their composition will be equal to the composition of their images. The concept of [[functoriality]] is essential in category theory as it allows us to study and analyze relationships between categories through the lens of functors. It provides a powerful tool for understanding structures and properties across different mathematical domains. # References ```dataview Table title as Title, authors as Authors where contains(subject, "functorial") sort title, authors, modified ```