Parametrised Representability
Often we define functors F where each object F(A) satisfies a universal property; these could be products, limits, colimits, (co)ends, (co)free objects, kan extensions, exponential objects... Usually, existence and uniqueness of the mediating morphisms allow you to define the functorial action and verify functoriality, respectively. But as these constructions get more complicated, the diagram-chasing gets more and more tedious, and one gets the feeling that there should be some universal theorem giving functoriality for free.
This talk will show such a silver bullet _does_ exist! In these situations of a **parametrised family of universal properties** there is an economical, "maximally lazy" way to define your functors on objects, and deduce the action on morphisms.
Licensed to the public under https://creativecommons.org/licenses/by/4.0/