equality (definitional?, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
A natural isomorphism $\eta\colon F \Rightarrow G$ between two functors $F$ and $G$
is equivalently
a natural transformation with a two-sided inverse;
a natural transformation each of whose components $\eta_c : F(c) \to G(c)$ for all $c \in Obj(C)$ is an isomorphism in $D$;
an isomorphism in the functor category $[C,D]$.
In this case, we say that $F$ and $G$ are naturally isomorphic.
If you want to speak of ‘the?’ functor satisfying certain conditions, then it should be unique up to unique natural isomorphism.
A natural isomorphism from a functor to itself is also called a natural automorphism.
natural isomorphism