Say a functor is "well-defined up to a self-equivalence of the source category" by certain properties/definition/construction iff, well, for any two functors $H,H':S\longrightarrow T$ with satisfying these properties/definition/obtained by this construction, there exists a self-equivalence of $s:S \longrightarrow S$ such that functors $H$ and $H'\circ s$ are equivalent.

Is there a nice way to reformulate this property "a functor unique up to self-equivalence of the source category", say in the language of 2-categories?

Are there any interesting examples of properties/definitions/constructions NOT involving arbitrary choice and yet such that the functor is well-defined up to a self-equivalence of source category ?

I am mostly interested to see an "algebraic" definition of a functor between "algebraic" categories which is well-defined up to self-equivalence but not well-defined.