Call two functors two functors $H,H':S\longrightarrow T$  _weakly equivalent_, or _equivalent up to a self-equivalence of the source category_, iff
there exists a self-equivalence of $s:S \longrightarrow S$ such that functors $H$ and $H'\circ s$ are equivalent.

>Can this property "weakly equivalent" be nicely reformulated in the language of higher category theory? or maybe homotopy theory?  
 
>Are there theorems claiming that any two functors with certain properties (not involving choice) are weakly equivalent?


That is, for such a theorem to be interesting, the properties should not explicitly involve an arbitrary choice, e.g. it should not say: choose a topology, bijection, self-equivalence and then construct the functor in the following way. Rather, a functor should be described in terms of preserving some structure etc. 


Below is the original question which was phrased very confusingly, it seems. I hope now it maybe is clearer. 

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.