You have two categories $C_1$ and $C_2$. We call a map of the classes $\mathrm{Ob}(C_1)\rightarrow \mathrm{Ob}(C_2)$ *a construction*. Sometimes you can find a functor $C_1\rightarrow C_2$ inducing this map, then you call your construction functorical or canonical.

Let us all a construction *very canonical* if there is a functor inducing it and between any two such functors there exists a natural isomorphism. What are some examples of very canonical constructions naturally arising in mathematics?

automorphismsof objects, which is strange. Also, uniqueness up to non-unique isomorphism is an unusual thing. $\endgroup$ – Mike Shulman Jun 2 at 16:00